Calculus In Exercises 7784, find the orthogonalprojection

Showing That a Function Is an Inner Product In Exercises 14, show that the function defines an inner product on R2, where u = (u1, u2) and v = (v1, v2). u, v = 3u1v1 + u2v2

Showing That a Function Is an Inner Product In Exercises 14, show that the function defines an inner product on R2, where u = (u1, u2) and v = (v1, v2). u, v = u1v1 + 9u2v2

Showing That a Function Is an Inner Product In Exercises 14, show that the function defines an inner product on R2, where u = (u1, u2) and v = (v1, v2). u, v = 1 2 u1v1 + 1 4 u2v2

Showing That a Function Is an Inner Product In Exercises 14, show that the function defines an inner product on R2, where u = (u1, u2) and v = (v1, v2). u, v = 2u1v2 + u2v1 + u1v2 + 2u2v2

Showing That a Function Is an Inner Product In Exercises 58, show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3). u, v = 2u1v1 + 3u2v2 + u3v3

Showing That a Function Is an Inner Product In Exercises 58, show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3). u, v = u1v1 + 2u2v2 + u3v3

Showing That a Function Is an Inner Product In Exercises 58, show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3). u, v = 4u1v1 + 3u2v2 + 2u3v3

Showing That a Function Is an Inner Product In Exercises 58, show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3). u, v = 1 2 u1v1 + 1 4 u2v2 + 1 2 u3v3

Showing That a Function Is Not an Inner Product In Exercises 912, show that the function does not define an inner product on R3, where u = (u1, u2) and v = (v1, v2). u, v = u1v1

Showing That a Function Is Not an Inner Product In Exercises 912, show that the function does not define an inner product on R3, where u = (u1, u2) and v = (v1, v2). u, v = u1v1 6u2v2

Showing That a Function Is Not an Inner Product In Exercises 912, show that the function does not define an inner product on R3, where u = (u1, u2) and v = (v1, v2). u, v = u1 2 v1 2 u2 2 v2 2

Showing That a Function Is Not an Inner Product In Exercises 912, show that the function does not define an inner product on R3, where u = (u1, u2) and v = (v1, v2). u, v = 3u1v2 u2v1

Showing That a Function Is Not an Inner Product In Exercises 1316, show that the function does not define an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3). u, v = u1u2u3

Showing That a Function Is Not an Inner Product In Exercises 1316, show that the function does not define an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3). u, v = u1v1 u2v2 u3v3

Showing That a Function Is Not an Inner Product In Exercises 1316, show that the function does not define an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3). u, v = u1 2 v1 2 + u2 2 v2 2 + u3 2 v2 2

Showing That a Function Is Not an Inner Product In Exercises 1316, show that the function does not define an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3). u, v = 2u1u2 + 3v1v2 + u3v3

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (3, 4), v = (5, 12), u, v = u v

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (1, 1), v = (6, 8), u, v = u v

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (4, 3), v = (0, 5), u, v = 3u1v1 + u2v2

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (0, 6), v = (1, 1), u, v = u1v1 + 2u2v2

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (0, 7, 2), v = (9, 3, 2), u, v = u v

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (0, 1, 2), v = (1, 2, 0), u, v = u v

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (8, 0, 8), v = (8, 3, 16), u, v = 2u1v1 + 3u2v2 + u3v3

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (1, 1, 1), v = (2, 5, 2), u, v = u1v1 + 2u2v2 + u3v3

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (1, 2, 0, 1), v = (0, 1, 2, 2), u, v = u v

Finding Inner Product, Length, and Distance In Exercises 1726, find (a) u, v, (b) )u), (c) )v), and (d) d(u, v) for the given inner product defined on Rn. u = (1, 1, 2, 0), v = (2, 1, 0, 1), u, v = u v

Showing That a Function Is an Inner Product In Exercises 27 and 28, let A = [ a11 a21 a12 a22] and B = [ b11 b21 b12 b22] be matrices in the vector space M2,2. Show that the function defines an inner product on M2,2. A, B = a11b11 + a12b12 + a21b21 + a22b22

Showing That a Function Is an Inner Product In Exercises 27 and 28, let A = [ a11 a21 a12 a22] and B = [ b11 b21 b12 b22] be matrices in the vector space M2,2. Show that the function defines an inner product on M2,2. A, B = 2a11b11 + a12b12 + a21b21 + 2a22b22

Finding Inner Product, Length, and Distance In Exercises 2932, find (a) A, B, (b) )A), (c) )B), and (d) d(A, B) for the matrices in M2,2 using the inner product A, B = 2a11b11 + a12b12 + a21b21 + 2a22b22. A = [ 2 3 4 1], B = [ 2 1 1 0]

Finding Inner Product, Length, and Distance In Exercises 2932, find (a) A, B, (b) )A), (c) )B), and (d) d(A, B) for the matrices in M2,2 using the inner product A, B = 2a11b11 + a12b12 + a21b21 + 2a22b22. A = [ 1 0 0 1], B = [ 0 1 1 0]

Finding Inner Product, Length, and Distance In Exercises 2932, find (a) A, B, (b) )A), (c) )B), and (d) d(A, B) for the matrices in M2,2 using the inner product A, B = 2a11b11 + a12b12 + a21b21 + 2a22b22. A = [ 1 2 1 4], B = [ 0 2 1 0]

Finding Inner Product, Length, and Distance In Exercises 2932, find (a) A, B, (b) )A), (c) )B), and (d) d(A, B) for the matrices in M2,2 using the inner product A, B = 2a11b11 + a12b12 + a21b21 + 2a22b22. A = [ 1 0 0 1], B = [ 1 0 1 1]

Showing That a Function Is an Inner Product In Exercises 33 and 34, show that the function defines an inner product for polynomials p(x) = a0 + a1x + . . . + anxn and q(x) = b0 + b1x + . . . + bnxn. p, q = a0b0 + 2a1b1 + a2b2 in P2

Showing That a Function Is an Inner Product In Exercises 33 and 34, show that the function defines an inner product for polynomials p(x) = a0 + a1x + . . . + anxn and q(x) = b0 + b1x + . . . + bnxn. p, q = a0b0 + a1b1 + . . . + anbn in Pn

Finding Inner Product, Length, and Distance In Exercises 3538, find (a) p, q, (b) ) p), (c) )q), and (d) d( p, q) for the polynomials in P2 using the inner product p, q = a0 b0 + a1b1 + a2b2. p(x) = 1 x + 3x2, q(x) = x x2

Finding Inner Product, Length, and Distance In Exercises 3538, find (a) p, q, (b) ) p), (c) )q), and (d) d( p, q) for the polynomials in P2 using the inner product p, q = a0 b0 + a1b1 + a2b2. p(x) = 1 + x + 1 2 x2, q(x) = 1 + 2x2

Finding Inner Product, Length, and Distance In Exercises 3538, find (a) p, q, (b) ) p), (c) )q), and (d) d( p, q) for the polynomials in P2 using the inner product p, q = a0 b0 + a1b1 + a2b2. p(x) = 1 + x2, q(x) = 1 x2

Finding Inner Product, Length, and Distance In Exercises 3538, find (a) p, q, (b) ) p), (c) )q), and (d) d( p, q) for the polynomials in P2 using the inner product p, q = a0 b0 + a1b1 + a2b2. p(x) = 1 3x + x2, q(x) = x + 2x2

Calculus In Exercises 3942, use the functions f and g in C[1, 1] to find (a) f, g, (b) ) f ), (c) )g), and (d) d( f, g) for the inner product f, g = 1 1 f(x)g(x) dx. f(x) = 1, g(x) = 4x2 1

Calculus In Exercises 3942, use the functions f and g in C[1, 1] to find (a) f, g, (b) ) f ), (c) )g), and (d) d( f, g) for the inner product f, g = 1 1 f(x)g(x) dx. f(x) = x, g(x) = x2 x + 2

Calculus In Exercises 3942, use the functions f and g in C[1, 1] to find (a) f, g, (b) ) f ), (c) )g), and (d) d( f, g) for the inner product f, g = 1 1 f(x)g(x) dx. f(x) = x, g(x) = ex

Calculus In Exercises 3942, use the functions f and g in C[1, 1] to find (a) f, g, (b) ) f ), (c) )g), and (d) d( f, g) for the inner product f, g = 1 1 f(x)g(x) dx. f(x) = x, g(x) = ex

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. u = (3, 4), v = (5, 12), u, v = u v

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. u = (3, 1), v = ( 1 3, 1), u, v = u v

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. u = (4, 3), v = (0, 5), u, v = 3u1v1 + u2v2

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. . u = ( 1 4, 1), v = (2, 1), u, v = 2u1v1 + u2v2

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. . u = (1, 1, 1), v = (2, 2, 2), u, v = u1v1 + 2u2v2 + u3v3

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. u = (0, 1, 2), v = (3, 2, 1), u, v = u v

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. p(x) = 1 x + x2, q(x) = 1 + x + x2, p, q = a0b0 + a1b1 + a2b2

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. p(x) = 1 + x2, q(x) = x x2, p, q = a0b0 + 2a1b1 + a2b2

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. Calculus f(x) = x, g(x) = x2, f, g = 1 1 f(x)g(x) dx

Finding the Angle Between Two Vectors In Exercises 4352, find the angle between the vectors. Calculus f(x) = 1, g(x) = x2, f, g = 1 1 f(x)g(x) dx

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. u = (5, 12), v = (3, 4), u, v = u v

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. u = (1, 1), v = (1, 1), u, v = u v

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. u = (0, 1, 5), v = (4, 3, 3), u, v = u v

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. u = (1, 0, 2), v = (1, 2, 0), u, v = u v

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. p(x) = 2x, q(x) = 1 + 3x2, p, q = a0b0 + a1b1 + a2b2

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. . p(x) = x, q(x) = 1 x2, p, q = a0b0 + 2a1b1 + a2b2

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. A = [ 0 2 3 1], B = [ 3 4 1 3], A, B = a11b11 + a12b12 + a21b21 + a22b22

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. A = [ 0 2 1 1], B = [ 1 2 1 2], A, B = a11b11 + a12b12 + a21b21 + a22b22

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. Calculus f(x) = sin x, g(x) = cos x, f, g = *4 0 f(x)g(x) dx

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. Calculus f(x) = x, g(x) = cos x, f, g = 2 0 f(x)g(x) dx

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. Calculus f(x) = x, g(x) = ex , f, g = 1 0 f(x)g(x) dx

Verifying Inequalities In Exercises 5364, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. Calculus f(x) = x, g(x) = ex , f, g = 1 0 f(x)g(x) dx

Calculus In Exercises 6568, show that f and g are orthogonal in the inner product space C[a, b] with the inner product f, g = b a f(x)g(x) dx. C[*2, *2], f(x) = cos x, g(x) = sin x

Calculus In Exercises 6568, show that f and g are orthogonal in the inner product space C[a, b] with the inner product f, g = b a f(x)g(x) dx. C[1, 1], f(x) = x, g(x) = 1 2(3x2 1)

Calculus In Exercises 6568, show that f and g are orthogonal in the inner product space C[a, b] with the inner product f, g = b a f(x)g(x) dx. C[1, 1], f(x) = x, g(x) = 1 2(5x3 3x

Calculus In Exercises 6568, show that f and g are orthogonal in the inner product space C[a, b] with the inner product f, g = b a f(x)g(x) dx. C[0, ], f(x) = 1, g(x) = cos(2nx), n = 1, 2, 3, . . .

Finding and Graphing Orthogonal Projections in R2 In Exercises 6972, (a) find projvu, (b) find projuv, and (c) sketch a graph of both projvu and projuv. Use the Euclidean inner product. u = (1, 2), v = (2, 1)

Finding and Graphing Orthogonal Projections in R2 In Exercises 6972, (a) find projvu, (b) find projuv, and (c) sketch a graph of both projvu and projuv. Use the Euclidean inner product. u = (3, 1), v = (6, 3)

Finding and Graphing Orthogonal Projections in R2 In Exercises 6972, (a) find projvu, (b) find projuv, and (c) sketch a graph of both projvu and projuv. Use the Euclidean inner product. u = (1, 3), v = (4, 4)

Finding and Graphing Orthogonal Projections in R2 In Exercises 6972, (a) find projvu, (b) find projuv, and (c) sketch a graph of both projvu and projuv. Use the Euclidean inner product. u = (2, 2), v = (3, 1)

Finding Orthogonal Projections In Exercises 7376, find (a) projvu and (b) projuv. Use the Euclidean inner product. u = (5, 3, 1), v = (1, 1, 0)

Finding Orthogonal Projections In Exercises 7376, find (a) projvu and (b) projuv. Use the Euclidean inner product. u = (1, 2, 1), v = (1, 2, 1)

Finding Orthogonal Projections In Exercises 7376, find (a) projvu and (b) projuv. Use the Euclidean inner product. u = (0, 1, 3, 6), v = (1, 1, 2, 2)

Finding Orthogonal Projections In Exercises 7376, find (a) projvu and (b) projuv. Use the Euclidean inner product. u = (1, 4, 2, 3), v = (2, 1, 2, 1)

Calculus In Exercises 7784, find the orthogonal projection of f onto g. Use the inner product in C[a, b] f, g = b a f(x)g(x) dx. C[1, 1], f(x) = x, g(x) = 1

Calculus In Exercises 7784, find the orthogonal projection of f onto g. Use the inner product in C[a, b] f, g = b a f(x)g(x) dx. C[1, 1], f(x) = x3 x, g(x) = 2x 1

Calculus In Exercises 7784, find the orthogonal projection of f onto g. Use the inner product in C[a, b] f, g = b a f(x)g(x) dx. C[0, 1], f(x) = x, g(x) = ex

Calculus In Exercises 7784, find the orthogonal projection of f onto g. Use the inner product in C[a, b] f, g = b a f(x)g(x) dx. C[0, 1], f(x) = x, g(x) = ex

Calculus In Exercises 7784, find the orthogonal projection of f onto g. Use the inner product in C[a, b] f, g = b a f(x)g(x) dx. C[, ], f(x) = sin x, g(x) = cos x

Calculus In Exercises 7784, find the orthogonal projection of f onto g. Use the inner product in C[a, b] f, g = b a f(x)g(x) dx. C[, ], f(x) = sin 2x, g(x) = cos 2x

Calculus In Exercises 7784, find the orthogonal projection of f onto g. Use the inner product in C[a, b] f, g = b a f(x)g(x) dx. C[, ], f(x) = x, g(x) = sin 2x

Calculus In Exercises 7784, find the orthogonal projection of f onto g. Use the inner product in C[a, b] f, g = b a f(x)g(x) dx. C[, ], f(x) = x, g(x) = cos 2x

True or False? In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The dot product is the only inner product that can be defined in Rn. (b) A nonzero vector in an inner product can have a norm of zero.

True or False? In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The norm of the vector u is the angle between u and the positive x-axis. (b) The angle between a vector v and the projection of u onto v is obtuse when the scalar a < 0 and acute when a > 0, where av = projvu.

Let u = (4, 2) and v = (2, 2) be vectors in R2 with the inner product u, v = u1v1 + 2u2v2. (a) Show that u and v are orthogonal. (b) Sketch u and v. Are they orthogonal in the Euclidean sense?

Proof Prove that )u + v)2 + )u v)2 = 2)u)2 + 2)v)2 for any vectors u and v in an inner product space V.

Proof Prove that the function is an inner product on Rn. u, v = c1u1v1 + c2u2v2 + . . . + cnunvn, ci > 0

Proof Let u and v be nonzero vectors in an inner product space V. Prove that u projvu is orthogonal to v.

Proof Prove Property 2 of Theorem 5.7: If u, v, and w are vectors in an inner product space V, then u + v, w = u, w + v, w.

Proof Prove Property 3 of Theorem 5.7: If u and v are vectors in an inner product space V and c is any real number, then u, cv = cu, v.

Guided Proof Let W be a subspace of the inner product space V. Prove that the set W = {v V: v, w = 0 for all w W} is a subspace of V. Getting Started: To prove that W is a subspace of V, you must show that W is nonempty and that the closure conditions for a subspace hold (Theorem 4.5). (i) Find a vector in W to conclude that it is nonempty. (ii) To show the closure of W under addition, you need to show that v1 + v2, w = 0 for all w W and for any v1, v2 W. Use the properties of inner products and the fact that v1, w and v2, w are both zero to show this. (iii) To show closure under multiplication by a scalar, proceed as in part (ii). Use the properties of inner products and the condition of belonging to W.

Use the result of Exercise 93 to find W when W is the span of (1, 2, 3) in V = R3.

Guided Proof Let u, v be the Euclidean inner product on Rn. Use the fact that u, v = uTv to prove that for any n n matrix A, (a) ATAu, v = u, Av and (b) ATAu, u = )Au)2. Getting Started: To prove (a) and (b), make use of both the properties of transposes (Theorem 2.6) and the properties of the dot product (Theorem 5.3). (i) To prove part (a), make repeated use of the property u, v = uTv and Property 4 of Theorem 2.6. (ii) To prove part (b), make use of the property u, v = uTv, Property 4 of Theorem 2.6, and Property 4 of Theorem 5.3.

CAPSTONE (a) Explain how to determine whether a function defines an inner product. (b) Let u and v be vectors in an inner product space V, such that v 0. Explain how to find the orthogonal projection of u onto v.

Finding Inner Product Weights In Exercises 97100, find c1 and c2 for the inner product of R2, u, v = c1u1v1 + c2u2v2 such that the graph represents a unit circle as shown. 2 3 ||u|| = 1

Finding Inner Product Weights In Exercises 97100, find c1 and c2 for the inner product of R2, u, v = c1u1v1 + c2u2v2 such that the graph represents a unit circle as shown. 1 3 13 1 4 1

Finding Inner Product Weights In Exercises 97100, find c1 and c2 for the inner product of R2, u, v = c1u1v1 + c2u2v2 such that the graph represents a unit circle as shown. ||u|| = 1 5 3 135

Finding Inner Product Weights In Exercises 97100, find c1 and c2 for the inner product of R2, u, v = c1u1v1 + c2u2v2 such that the graph represents a unit circle as shown. 6 6 4 6 4 6

Consider the vectors u = (6, 2, 4) and v = (1, 2, 0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the vector closest to uthat is, show that d(u, projvu) is a minimum.

Showing That a Function Is an Inner Product In Exercises 1 - 4, show that the function defines an inner product on \(R^{2}\), where \(\mathrm{u}=\left(u_{1}, u_{2}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=3 u_{1} v_{1}+u_{2} v_{2}\) Text Transcription: R^2 u = (u_1, u_2) v = (v_1, v_2) langle u, v rangle = 3u_1 v_1 + u_2 v_2

Showing That a Function Is an Inner Product In Exercises 1 - 4, show that the function defines an inner product on \(R^{2}\), where \(\mathrm{u}=\left(u_{1}, u_{2}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+9 u_{2} v_{2}\) Text Transcription: R^2 u = (u_1, u_2) v = (v_1, v_2) langle u, v rangle = u_1 v_1 + 9u_2v_2

Showing That a Function Is an Inner Product In Exercises 1 - 4, show that the function defines an inner product on \(R^{2}\), where \(\mathrm{u}=\left(u_{1}, u_{2}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=\frac{1}{2} u_{1} v_{1}+\frac{1}{4} u_{2} v_{2}\) Text Transcription: R^2 u = (u_1, u_2) v = (v_1, v_2) langle u, v rangle = 1 / 2 u_1 v_1 + 1 / 4 u_2 v_2

Showing That a Function Is an Inner Product In Exercises 1 - 4, show that the function defines an inner product on \(R^{2}\), where \(\mathrm{u}=\left(u_{1}, u_{2}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=2 u_{1} v_{2}+u_{2} v_{1}+u_{1} v_{2}+2 u_{2} v_{2}\) Text Transcription: R^2 u = (u_1, u_2) v = (v_1, v_2) langle u, v rangle = 2u_1 v_2 + u_2 v_1 + u_1 v_2 + 2u_2 v_2

Showing That a Function Is an Inner Product In Exercises 5 - 8, show that the function defines an inner product on \(R^{3}\), where \(u=\left(u_{1}, u_{2}, u_{3}\right)\) and \(v=\left(v_{1}, v_{2}, v_{3}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=2 u_{1} v_{1}+3 u_{2} v_{2}+u_{3} v_{3}\) Text Transcription: R^3 u = (u_1, u_2, u_3) v = (v_1, v_2, v_3) langle u, v rangle = 2u_1 v_1 + 3u_2 v_2 + u_3 v_3

Showing That a Function Is an Inner Product In Exercises 5 - 8, show that the function defines an inner product on \(R^{3}\), where \(u=\left(u_{1}, u_{2}, u_{3}\right)\) and \(v=\left(v_{1}, v_{2}, v_{3}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+2 u_{2} v_{2}+u_{3} v_{3}\) Text Transcription: R^3 u = (u_1, u_2, u_3) v = (v_1, v_2, v_3) langle u, v rangle = u_1 v_1 + 2u_2 v_2 + u_3 v_3

Showing That a Function Is an Inner Product In Exercises 5 - 8, show that the function defines an inner product on \(R^{3}\), where \(u=\left(u_{1}, u_{2}, u_{3}\right)\) and \(v=\left(v_{1}, v_{2}, v_{3}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=4 u_{1} v_{1}+3 u_{2} v_{2}+2 u_{3} v_{3}\) Text Transcription: R^3 u = (u_1, u_2, u_3) v = (v_1, v_2, v_3) langle u, v rangle = 4u_1 v_1 + 3u_2 v_2 + 2u_3 v_3

Showing That a Function Is an Inner Product In Exercises 5 - 8, show that the function defines an inner product on \(R^{3}\), where \(u=\left(u_{1}, u_{2}, u_{3}\right)\) and \(v=\left(v_{1}, v_{2}, v_{3}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=\frac{1}{2} u_{1} v_{1}+\frac{1}{4} u_{2} v_{2}+\frac{1}{2} u_{3} v_{3}\) Text Transcription: R^3 u = (u_1, u_2, u_3) v = (v_1, v_2, v_3) langle u, v rangle = 1 / 2 u_1 v_1 + 1 / 4 u_2 v_2 + 1 / 2 u_3 v_3

Showing That a Function Is Not an Inner Product In Exercises 9 - 12, show that the function does not define an inner product on \(R^{3}\), where \(\mathrm{u}=\left(u_{1}, u_{2}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}\) Text Transcription: R^3 u = (u_1, u_2) v = (v_1, v_2) langle u, v rangle = u_1 v_1

Showing That a Function Is Not an Inner Product In Exercises 9 - 12, show that the function does not define an inner product on \(R^{3}\), where \(\mathrm{u}=\left(u_{1}, u_{2}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}-6 u_{2} v_{2}\) Text Transcription: R^3 u = (u_1, u_2) v = (v_1, v_2) langle u, v rangle = u_1 v_1 - 6u_2 v_2

Showing That a Function Is Not an Inner Product In Exercises 9 - 12, show that the function does not define an inner product on \(R^{3}\), where \(\mathrm{u}=\left(u_{1}, u_{2}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1}^{2} v_{1}^{2}-u_{2}^{2} v_{2}^{2}\) Text Transcription: R^3 u = (u_1, u_2) v = (v_1, v_2) langle u, v rangle = u_1^2 v_1^2 - u_2^2 v_2^2

Showing That a Function Is Not an Inner Product In Exercises 9 - 12, show that the function does not define an inner product on \(R^{3}\), where \(\mathrm{u}=\left(u_{1}, u_{2}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}\right)\). \(\langle\mathbf{u}, \mathbf{v}\rangle=3 u_{1} v_{2}-u_{2} v_{1}\) Text Transcription: R^3 u = (u_1, u_2) v = (v_1, v_2) langle u, v rangle = 3u_1 v_2 - u_2 v_1

Showing That a Function Is Not an Inner Product In Exercises 13-16, show that the function does not define an inner product on \(R^{3}\), where \(\mathrm{u}=\left(u_{1}, u_{2}, u_{3}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}, v_{3}\right)\) \(\langle\mathbf{u}, \mathbf{v}\rangle=-u_{1} u_{2} u_{3}\) Text Transcription: R^3 u = (u_1, u_2, u_3) v = (v_1, v_2, v_3) langle u, v rangle = -u_1 u_2 u_3

Showing That a Function Is Not an Inner Product In Exercises 13-16, show that the function does not define an inner product on \(R^{3}\), where \(\mathrm{u}=\left(u_{1}, u_{2}, u_{3}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}, v_{3}\right)\) \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}-u_{2} v_{2}-u_{3} v_{3}\) Text Transcription: R^3 u = (u_1, u_2, u_3) v = (v_1, v_2, v_3) langle u, v rangle = u_1 v_1 - u_2 v_2 - u_3 v_3

Showing That a Function Is Not an Inner Product In Exercises 13-16, show that the function does not define an inner product on \(R^{3}\), where \(\mathrm{u}=\left(u_{1}, u_{2}, u_{3}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}, v_{3}\right)\) \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1}^{2} v_{1}^{2}+u_{2}^{2} v_{2}^{2}+u_{3}^{2} v_{2}^{2}\) Text Transcription: R^3 u = (u_1, u_2, u_3) v = (v_1, v_2, v_3) langle u, v rangle = u_1^2 v_1^2 + u_2^2 v_2^2 + u_3^2 v_2^2

Showing That a Function Is Not an Inner Product In Exercises 13-16, show that the function does not define an inner product on \(R^{3}\), where \(\mathrm{u}=\left(u_{1}, u_{2}, u_{3}\right)\) and \(\mathrm{v}=\left(v_{1}, v_{2}, v_{3}\right)\) \(\langle\mathbf{u}, \mathbf{v}\rangle=2 u_{1} u_{2}+3 v_{1} v_{2}+u_{3} v_{3}\) Text Transcription: R^3 u = (u_1, u_2, u_3) v = (v_1, v_2, v_3) langle u, v rangle = 2 u_1 u_2 + 3 v_1 v_2 + u_3 v_3

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(3,4), \quad \mathbf{v}=(5,-12), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (3, 4), v = (5, -12), langle u, v rangle = u cdot v

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(-1,1), \quad \mathbf{v}=(6,8), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (-1,1), v = (6, 8), langle u, v rangle = u cdot v

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(-4,3), \quad \mathbf{v}=(0,5), \quad\langle\mathbf{u}, \mathbf{v}\rangle=3 u_{1} v_{1}+u_{2} v_{2}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (-4, 3), v = (0, 5), langle u, v rangle = 3 u_1 v_1 + u_2 v_2

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(0,-6), \quad \mathbf{v}=(-1,1), \quad\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+2 u_{2} v_{2}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (0, -6), v = (-1, 1), langle u, v rangle = u_1 v_1 + 2 u_2 v_2

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(0,7,2), \quad \mathbf{v}=(9,-3,-2), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (0, 7, 2), v = (9, -3, -2), langle u, v rangle = u cdot v

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(0,1,2), \quad \mathbf{v}=(1,2,0), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (0, 1, 2), v = (1, 2, 0), langle u, v rangle = u cdot v

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(8,0,-8), \quad \mathbf{v}=(8,3,16)\), \(\langle\mathbf{u}, \mathbf{v}\rangle=2 u_{1} v_{1}+3 u_{2} v_{2}+u_{3} v_{3}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (8, 0, -8), v = (8, 3, 16) langle u, v rangle = 2u_1 v_1 + 3 u_2 v_2 + u_3 v_3

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(1,1,1), \quad \mathbf{v}=(2,5,2)\), \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+2 u_{2} v_{2}+u_{3} v_{3}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (1, 1, 1), v = (2, 5, 2) langle u, v rangle = u_1 v_1 + 2 u_2 v_2 + u_3 v_3

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(-1,2,0,1), \quad \mathbf{v}=(0,1,2,2), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (-1, 2, 0,1), v = (0, 1, 2, 2), langle u, v rangle = u cdot v

Finding Inner Product, Length, and Distance In Exercises 17 - 26, find (a) \(\langle\mathbf{u}, \mathbf{v}\rangle\), (b) \(\|\mathbf{u}\|\), (c) \(\|\mathbf{v}\|\), and (d) \(d(\mathrm{u}, \mathrm{v})\) for the given inner product defined on \(R^{n}\). \(\mathbf{u}=(1,-1,2,0), \quad \mathbf{v}=(2,1,0,-1), \quad \langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: langle u, v rangle ||u}|| ||v|| d (u, v) R^n u = (1, -1, 2, 0), v = (2, 1, 0, -1), langle u, v rangle = u cdot v

Showing That a Function Is an Inner Product In Exercises 27 and 28, let \(A=\left[\begin{array}{ll} a_{11} a_{12} \\ a_{21} a_{22} \end{array}\right]\) and \(B=\left[\begin{array}{ll} b_{11} b_{12} \\ b_{21} b_{22} \end{array}\right]\) be matrices in the vector space \(M_{2,2}\). Show that the function defines an inner product on \(M_{2,2}\). \(\langle A, B\rangle=a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+a_{22} b_{22}\) Text Transcription: A = [a_11 a_12 \\ a_21 a_22] B = [b_11 b_12 \\ b_21 b_22] M_{2, 2} langle A, B rangle = a_11 b_11 + a_12 b_12 + a_21 b_21 + a_22 b_22

Showing That a Function Is an Inner Product In Exercises 27 and 28 , let \(A=\left[\begin{array}{ll} a_{11} a_{12} \\ a_{21} a_{22} \end{array}\right]\) and \(B=\left[\begin{array}{ll} b_{11} b_{12} \\ b_{21} b_{22} \end{array}\right]\) be matrices in the vector space \(M_{2,2}\). Show that the function defines an inner product on \(M_{2,2}\). \(\langle A, B\rangle=2 a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+2 a_{22} b_{22}\) Text Transcription: A = [a_11 a_12 \\ a_21 a_22] B = [b_11 b_12 \\ b_21 b_22] M_{2, 2} langle A, B rangle = 2a_11 b_11 + a_12 b_12 + a_21 b_21 + 2a_22 b_22

Finding Inner Product, Length, and Distance In Exercises 29 - 32, find (a) \(\langle A, B \rangle\), (b) \(||A\||\), (c) \(||B||\), and (d) \(d(A, B)\) for the matrices in \(M_{2,2}\) using the inner product \(\langle A, B\rangle=2 a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+2 a_{22} b_{22}\). \(A=\left[\begin{array}{rr}2 -4 \\ -3 1\end{array}\right], \quad B=\left[\begin{array}{rr}-2 1 \\ 1 0\end{array}\right]\) Text Transcription: langle A, B rangle ||A\|| ||B|| d(A, B) M_{2,2} langle A, Brangle = 2 a_11 b_11 + a_12 b_12 + a_21 b_21 + 2 a_22 b_22 A = [2 -4 \\ -3 1], B = [-2 1 \\ 1 0]

Finding Inner Product, Length, and Distance In Exercises 29 - 32, find (a) \(\langle A, B \rangle\), (b) \(||A\||\), (c) \(||B||\), and (d) \(d(A, B)\) for the matrices in \(M_{2,2}\) using the inner product \(\langle A, B\rangle=2 a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+2 a_{22} b_{22}\) \(A=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right], \quad B=\left[\begin{array}{ll}0 1 \\ 1 0\end{array}\right]\) Text Transcription: langle A, B rangle ||A\|| ||B|| d(A, B) M_{2,2} langle A, Brangle = 2 a_11 b_11 + a_12 b_12 + a_21 b_21 + 2 a_22 b_22 A = [1 0 \\ 0 1], B = [0 1 \\ 1 0]

Finding Inner Product, Length, and Distance In Exercises 29 - 32, find (a) \(\langle A, B \rangle\), (b) \(||A\||\), (c) \(||B||\), and (d) \(d(A, B)\) for the matrices in \(M_{2,2}\) using the inner product \(\langle A, B\rangle=2 a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+2 a_{22} b_{22}\) \(A=\left[\begin{array}{rr}1 -1 \\ 2 4\end{array}\right], \quad B=\left[\begin{array}{rr}0 1 \\ -2 0\end{array}\right]\) Text Transcription: langle A, B rangle ||A\|| ||B|| d(A, B) M_{2,2} langle A, Brangle = 2 a_11 b_11 + a_12 b_12 + a_21 b_21 + 2 a_22 b_22 A = [1 -1 \\ 2 4], B = [0 1 \\ -2 0]

Finding Inner Product, Length, and Distance In Exercises 29 - 32, find (a) \(\langle A, B \rangle\), (b) \(||A\||\), (c) \(||B||\), and (d) \(d(A, B)\) for the matrices in \(M_{2,2}\) using the inner product \(\langle A, B\rangle=2 a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+2 a_{22} b_{22}\) \(A=\left[\begin{array}{rr}1 0 \\ 0 -1\end{array}\right], \quad B=\left[\begin{array}{rr}1 1 \\ 0 -1\end{array}\right]\) Text Transcription: langle A, B rangle ||A\|| ||B|| d(A, B) M_{2,2} langle A, Brangle = 2 a_11 b_11 + a_12 b_12 + a_21 b_21 + 2 a_22 b_22 A = [1 0 \\ 0 -1], B = [1 1 \\ 0 -1]

Showing That a Function Is an Inner Product In Exercises33 and34, show that the function defines an inner product for polynomials \(p(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n}\) and \(q(x)=b_{0}+b_{1} x+\cdots+b_{n} x^{n}\). \(\langle p, q\rangle=a_{0} b_{0}+2 a_{1} b_{1}+a_{2} b_{2}\) in \(P_{2}\) Text Transcription: p(x) = a_0 + a_1 x + cdots + a_n x^n q(x) = b_0 + b_1 x + cdots + b_n x^n langle p, q rangle = a_0 b_0 + 2 a_1 b_1 + a_2 b_2 P_2

Showing That a Function Is an Inner Product In Exercises33 and34, show that the function defines an inner product for polynomials \(p(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n}\) and \(q(x)=b_{0}+b_{1} x+\cdots+b_{n} x^{n}\). \(\langle p, q\rangle=a_{0} b_{0}+a_{1} b_{1}+\cdots+a_{n} b_{n}\) in \(P_{n}\) Text Transcription: p(x) = a_0 + a_1 x + cdots + a_n x^n q(x) = b_0 + b_1 x + cdots + b_n x^n langle p, q rangle = a_0 b_0 + a_1 b_1 + cdots + a_n b_n P_n

Finding Inner Product, Length, and Distance In Exercises 35 - 38, find (a) \(\langle p, q\rangle\), (b) \(||p||\), (c) \(||q||\), and (d) \(d(p, q)\) for the polynomials in \(P_{2}\) using the inner product \(\langle p, q\rangle=a_{0} b_{0}+a_{1} b_{1}+a_{2} b_{2}\) \(p(x)=1-x+3 x^{2}, \quad q(x)=x-x^{2}\) Text Transcription: langle p, q rangle ||p|| ||q|| d(p, q) P_2 langle p, q rangle = a_0 b_0 + a_1 b_1 + a_2 b_2 p(x) = 1 - x + 3x^2, q(x) = x - x^2

Finding Inner Product, Length, and Distance In Exercises 35 - 38, find (a) \(\langle p, q\rangle\), (b) \(||p||\), (c) \(||q||\), and (d) \(d(p, q)\) for the polynomials in \(P_{2}\) using the inner product \(\langle p, q\rangle=a_{0} b_{0}+a_{1} b_{1}+a_{2} b_{2}\) \(p(x)=1+x+\frac{1}{2} x^{2}, \quad q(x)=1+2 x^{2}\) Text Transcription: langle p, q rangle ||p|| ||q|| d(p, q) P_2 langle p, q rangle = a_0 b_0 + a_1 b_1 + a_2 b_2 p(x) = 1 + x + 1 / 2 x^2, q(x) = 1 + 2x^2

Finding Inner Product, Length, and Distance In Exercises 35 - 38, find (a) \(\langle p, q\rangle\), (b) \(||p||\), (c) \(||q||\), and (d) \(d(p, q)\) for the polynomials in \(P_{2}\) using the inner product \(\langle p, q\rangle=a_{0} b_{0}+a_{1} b_{1}+a_{2} b_{2}\) \(p(x)=1+x^{2}, \quad q(x)=1-x^{2}\) Text Transcription: langle p, q rangle ||p|| ||q|| d(p, q) P_2 langle p, q rangle = a_0 b_0 + a_1 b_1 + a_2 b_2 p(x) = 1 + x^2, q(x) = 1 - x^2

Finding Inner Product, Length, and Distance In Exercises 35 - 38, find (a) \(\langle p, q\rangle\), (b) \(||p||\), (c) \(||q||\), and (d) \(d(p, q)\) for the polynomials in \(P_{2}\) using the inner product \(\langle p, q\rangle=a_{0} b_{0}+a_{1} b_{1}+a_{2} b_{2}\) \(p(x)=1-3 x+x^{2}, \quad q(x)=-x+2 x^{2}\) Text Transcription: langle p, q rangle ||p|| ||q|| d(p, q) P_2 langle p, q rangle = a_0 b_0 + a_1 b_1 + a_2 b_2 p(x) = 1 - 3x + x^2, q(x) = -x + 2x^2

Calculus In Exercises 39 - 42, use the functions f and g in C[-1, 1] to find (a) \(\langle f, g\rangle\), (b) \(||f||\), (c) \(||g||\), and (d) \(d(f, g)\) for the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\) \(f(x)=1, \quad g(x)=4 x^{2}-1\) Text Transcription: langle f, g rangle ||f|| ||g|| d(f, g) langle f, g rangle = int_{-1}^{1} f(x) g(x) dx f(x) = 1, g(x) = 4x^2 - 1

Calculus In Exercises 39 - 42, use the functions f and g in C[-1, 1] to find (a) \(\langle f, g\rangle\), (b) \(||f||\), (c) \(||g||\), and (d) \(d(f, g)\) for the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\) \(f(x)=-x, \quad g(x)=x^{2}-x+2\) Text Transcription: langle f, g rangle ||f|| ||g|| d(f, g) langle f, g rangle = int_{-1}^{1} f(x) g(x) dx f(x) = -x, g(x) = x^2 - x + 2

Calculus In Exercises 39 - 42, use the functions f and g in C[-1, 1] to find (a) \(\langle f, g\rangle\), (b) \(||f||\), (c) \(||g||\), and (d) \(d(f, g)\) for the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\) \(f(x)=x, \quad g(x)=e^{x}\) Text Transcription: langle f, g rangle ||f|| ||g|| d(f, g) langle f, g rangle = int_{-1}^{1} f(x) g(x) dx f(x) = x, g(x) = e^x

Calculus In Exercises 39 - 42, use the functions f and g in C[-1, 1] to find (a) \(\langle f, g\rangle\), (b) \(||f||\), (c) \(||g||\), and (d) \(d(f, g)\) for the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\) \(f(x)=x, \quad g(x)=e^{-x}\) Text Transcription: langle f, g rangle ||f|| ||g|| d(f, g) langle f, g rangle = int_{-1}^{1} f(x) g(x) dx f(x) = x, g(x) = e^-x

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. \(\mathbf{u}=(3,4), \quad \mathbf{v}=(5,-12), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: theta u = (3, 4), v = (5, -12), langle u, v rangle = u cdot v

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. \(\mathbf{u}=(3,-1), \quad \mathbf{v}=\left(\frac{1}{3}, 1\right), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: theta u = (3,-1), v = (1 / 3, 1), langle u, v rangle = u cdot v

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. \(\mathbf{u}=(-4,3), \quad \mathbf{v}=(0,5), \quad\langle\mathbf{u}, \mathbf{v}\rangle=3 u_{1} v_{1}+u_{2} v_{2}\) Text Transcription: theta u = (-4, 3), v = (0,5), langle u, v rangle = 3 u_1 v_1 + u_2 v_2

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. \(\mathbf{u}=\left(\frac{1}{4},-1\right), \quad \mathbf{v}=(2,1)\), \(\langle\mathbf{u}, \mathbf{v}\rangle=2 u_{1} v_{1}+u_{2} v_{2}\) Text Transcription: theta u = (1 / 4, -1), v = (2, 1) langle u, v rangle = 2 u_{1} v_1+ u_2 v_2

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. \(\mathbf{u}=(1,1,1), \quad \mathbf{v}=(2,-2,2)\), \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+2 u_{2} v_{2}+u_{3} v_{3}\) Text Transcription: theta u = (1, 1, 1), v = (2, -2, 2) langle u, v rangle = u_1 v_1 + 2 u_2 v_2 + u_3 v_3

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. \(\mathbf{u}=(0,1,-2), \quad \mathbf{v}=(3,-2,1),\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: theta u = (0, 1, -2), v = (3, -2, 1), langle u, v rangle = u cdot v

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. \(p(x)=1-x+x^{2}, \quad q(x)=1+x+x^{2}\), \(\langle p, q\rangle=a_{0} b_{0}+a_{1} b_{1}+a_{2} b_{2}\) Text Transcription: theta p(x) = 1 - x + x^2, q(x) = 1 + x + x^2 langle p, q rangle = a_0 b_0 + a_1 b_1 + a_2 b_2

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. \(p(x)=1+x^{2}, \quad q(x)=x-x^{2}\), \(\langle p, q\rangle=a_{0} b_{0}+2 a_{1} b_{1}+a_{2} b_{2}\) Text Transcription: theta p(x) = 1 + x^2, q(x) = x - x^2 langle p, q rangle = a_0 b_0 + 2a_1 b_1 + a_2 b_2

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. Calculus \(f(x)=x, \quad g(x)=x^{2}\), \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\) Text Transcription: theta f(x) = x, g(x) = x^2 langle f, g rangle = int_{-1}^{1} f(x) g(x) dx

Finding the Angle Between Two Vectors In Exercises 43-52, find the angle \(\theta\) between the vectors. Calculus \(f(x)=1, \quad g(x)=x^{2}\), \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\) Text Transcription: theta f(x) = 1, g(x) = x^2 langle f, g rangle = int_{-1}^{1} f(x) g(x) dx

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. \(\mathbf{u}=(5,12), \quad \mathbf{v}=(3,4), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: u = (5, 12), v = (3, 4), langle u, v rangle = u cdot v

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. \(\mathbf{u}=(-1,1), \quad \mathbf{v}=(1,-1),\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: u = (-1,1), v = (1,-1), langle u, v rangle = u cdot v

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. \(\mathbf{u}=(0,1,5), \quad \mathbf{v}=(-4,3,3),\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: u = (0, 1, 5), v = (-4, 3, 3), langle u, v rangle = u cdot v

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. \(\mathbf{u}=(1,0,2), \quad \mathbf{v}=(1,2,0), \quad\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u} \cdot \mathbf{v}\) Text Transcription: u = (1, 0, 2), v = (1, 2, 0), langle u, v rangle = u cdot v

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. \(p(x)=2 x, \quad q(x)=1+3 x^{2}\), \(\langle p, q\rangle=a_{0} b_{0}+a_{1} b_{1}+a_{2} b_{2}\) Text Transcription: p(x) = 2x, q(x) = 1 + 3x^2 langle p, q rangle = a_0 b_0 + a_1 b_1 + a_2 b_2

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. \(p(x)=x, \quad q(x)=1-x^{2}\), \(\langle p, q\rangle=a_{0} b_{0}+2 a_{1} b_{1}+a_{2} b_{2}\) Text Transcription: p(x) = x, q(x) = 1 - x^2 langle p, q rangle = a_0 b_0 + 2 a_1 b_1 + a_2 b_2

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. \(A=\left[\begin{array}{ll}0 3 \\ 2 1\end{array}\right], \quad B=\left[\begin{array}{rr}-3 1 \\ 4 3\end{array}\right]\), \(\langle A, B\rangle=a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+a_{22} b_{22}\) Text Transcription: A = [0 3 \\ 2 1], B = [-3 1 \\ 4 3] langle A, Brangle = a_11 b_11 + a_12 b_12 + a_21 b_21 + a_22 b_22

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. \(A=\left[\begin{array}{rr}0 1 \\ 2 -1\end{array}\right], \quad B=\left[\begin{array}{rr}1 1 \\ 2 -2\end{array}\right]\) \(\langle A, B\rangle=a_{11} b_{11}+a_{12} b_{12}+a_{21} b_{21}+a_{22} b_{22}\) Text Transcription: A = [0 1 \\ 2 -1], B = [1 1 \\ 2 -2] langle A, B rangle = a_11 b_11 + a_12 b_12 + a_21 b_21 + a_22 b_22

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. Calculus \(f(x)=\sin x, \quad g(x)=\cos x\), \(\langle f, g\rangle=\int_{0}^{\pi / 4} f(x) g(x) d x\) Text Transcription: f(x) = sin x, g(x) = cos x langle f, g rangle = int_{0}^{pi/ 4} f(x) g(x) dx

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. Calculus \(f(x)=x, \quad g(x)=\cos \pi x\), \(\langle f, g\rangle=\int_{0}^{2} f(x) g(x) d x\) Text Transcription: f(x) = x, g(x) = cos pi x langle f, g rangle = int_0^2 f(x) g(x) dx

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. Calculus \(f(x)=x, \quad g(x)=e^{x}\), \(\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x\) Text Transcription: f(x) = x, g(x) = e^x langle f, g rangle = int_0^{1} f(x) g(x) dx

Verifying Inequalities In Exercises 53 - 64, verify (a) the Cauchy-Schwarz Inequality and (b) the triangle inequality for the given vectors and inner products. Calculus \(f(x)=x, \quad g(x)=e^{-x}\), \(\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x\) Text Transcription: f(x) = x, g(x) = e^-x langle f, g rangle = int_0^1 f(x) g(x) dx

Calculus In Exercises 65 - 68, show that f and g are orthogonal in the inner product space C[a, b] with the inner product \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[-\pi / 2, \pi / 2], \quad f(x)=\cos x, \quad g(x)=\sin x\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C [- pi/2, pi/2], f(x) = cos x, g(x) = sin x

Calculus In Exercises 65 - 68, show that f and g are orthogonal in the inner product space C[a, b] with the inner product \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[-1,1], \quad f(x)=x, \quad g(x)=\frac{1}{2}\left(3 x^{2}-1\right)\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[-1, 1], f(x) = x, g(x) = 1 / 2 (3x^2 - 1)

Calculus In Exercises 65 - 68, show that f and g are orthogonal in the inner product space C[a, b] with the inner product \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[-1,1], \quad f(x)=x, \quad g(x)=\frac{1}{2}\left(5 x^{3}-3 x\right)\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[-1, 1], f(x) = x, g(x) = 1 / 2 (5x^3 - 3x)

Calculus In Exercises 65 - 68, show that f and g are orthogonal in the inner product space C[a, b] with the inner product \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[0, \pi], \quad f(x)=1, \quad g(x)=\cos (2 n x)\), n = 1, 2, 3, . . . Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[0, pi], f(x) = 1, g(x) = cos (2nx)

Finding and Graphing Orthogonal Projections in \(R^{2}\) In Exercises 69 - 72, (a) find \(proj_{v} u\), (b) find \(proj_{u} v\), and (c) sketch a graph of both \(proj_{v} u\) and \(proj_{u} v\). Use the Euclidean inner product. \(\mathbf{u}=(1,2), \quad \mathbf{v}=(2,1)\) Text Transcription: R^2 proj_{v} u proj_{u} v u = (1, 2), v = (2, 1)

Finding and Graphing Orthogonal Projections in \(R^{2}\) In Exercises 69 - 72, (a) find \(proj_{v} u\), (b) find \(proj_{u} v\), and (c) sketch a graph of both \(proj_{v} u\) and \(proj_{u} v\). Use the Euclidean inner product. \(\mathbf{u}=(-3,-1), \quad \mathbf{v}=(6,3)\) Text Transcription: R^2 proj_{v} u proj_{u} v u = (-3, -1), v = (6, 3)

Finding and Graphing Orthogonal Projections in \(R^{2}\) In Exercises 69 - 72, (a) find \(proj_{v} u\), (b) find \(proj_{u} v\), and (c) sketch a graph of both \(proj_{v} u\) and \(proj_{u} v\). Use the Euclidean inner product. \(\mathbf{u}=(-1,3), \quad \mathbf{v}=(4,4)\) Text Transcription: R^2 proj_{v} u proj_{u} v u = (-1, 3), v = (4, 4)

Finding and Graphing Orthogonal Projections in \(R^{2}\) In Exercises 69 - 72, (a) find \(proj_{v} u\), (b) find \(proj_{u} v\), and (c) sketch a graph of both \(proj_{v} u\) and \(proj_{u} v\). Use the Euclidean inner product. \(\mathbf{u}=(2,-2), \quad \mathbf{v}=(3,1)\) Text Transcription: R^2 proj_{v} u proj_{u} v u = (2, -2), v = (3, 1)

Finding Orthogonal Projections In Exercises 73 - 76, find (a) \(proj_{v} u\) and (b) \(proj_{u} v\). Use the Euclidean inner product. \(\mathbf{u}=(5,-3,1), \quad \mathbf{v}=(1,-1,0)\) Text Transcription: proj_{v} u proj_{u} v u = (5, -3, 1), v = (1, -1, 0)

Finding Orthogonal Projections In Exercises 73 - 76, find (a) \(proj_{v} u\) and (b) \(proj_{u} v\). Use the Euclidean inner product. \(\mathbf{u}=(1,2,-1), \quad \mathbf{v}=(-1,2,-1)\) Text Transcription: proj_{v} u proj_{u} v u = (1, 2, -1), v = (-1, 2, -1)

Finding Orthogonal Projections In Exercises 73 - 76, find (a) \(proj_{v} u\) and (b) \(proj_{u} v\). Use the Euclidean inner product. \(\mathbf{u}=(0,1,3,-6), \quad \mathbf{v}=(-1,1,2,2)\) Text Transcription: proj_{v} u proj_{u} v u = (0, 1, 3, -6), v = (-1, 1, 2, 2)

Finding Orthogonal Projections In Exercises 73 - 76, find (a) \(proj_{v} u\) and (b) \(proj_{u} v\). Use the Euclidean inner product. \(\mathbf{u}=(-1,4,-2,3), \quad \mathbf{v}=(2,-1,2,-1)\) Text Transcription: proj_{v} u proj_{u} v u = (-1, 4, -2, 3), v = (2, -1, 2, -1)

Calculus In Exercises 77 - 84, find the orthogonal projection of f onto g. Use the inner product in C[a, b] \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) C[-1, 1], f(x) = x, g(x) = 1 Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx

Calculus In Exercises 77 - 84, find the orthogonal projection of f onto g. Use the inner product in C[a, b] \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[-1,1], \quad f(x)=x^{3}-x, \quad g(x)=2 x-1\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[-1, 1], f(x) = x^3 - x, g(x) = 2x - 1

Calculus In Exercises 77 - 84, find the orthogonal projection of f onto g. Use the inner product in C[a, b] \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[0,1], \quad f(x)=x, \quad g(x)=e^{x}\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[0, 1], f(x) = x, g(x) = e^x

Calculus In Exercises 77 - 84, find the orthogonal projection of f onto g. Use the inner product in C[a, b] \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[0,1], \quad f(x)=x, \quad g(x)=e^{-x}\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[0, 1], f(x) = x, g(x) = e^-x

Calculus In Exercises 77 - 84, find the orthogonal projection of f onto g. Use the inner product in C[a, b] \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[-\pi, \pi], \quad f(x)=\sin x, \quad g(x)=\cos x\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[-pi, pi], f(x) = sin x, g(x) = cos x

Calculus In Exercises 77 - 84, find the orthogonal projection of f onto g. Use the inner product in C[a, b] \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[-\pi, \pi], \quad f(x)=\sin 2 x, \quad g(x)=\cos 2 x\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[- pi, pi], f(x) = sin 2x, g(x) = cos 2x

Calculus In Exercises 77 - 84, find the orthogonal projection of f onto g. Use the inner product in C[a, b] \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[-\pi, \pi], \quad f(x)=x, \quad g(x)=\sin 2 x\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[- pi, pi], f(x) = x, g(x) = sin 2x

Calculus In Exercises 77 - 84, find the orthogonal projection of f onto g. Use the inner product in C[a, b] \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) d x\) \(C[-\pi, \pi], \quad f(x)=x, \quad g(x)=\cos 2 x\) Text Transcription: langle f, g rangle = int_a^b f(x) g(x) dx C[-pi, pi], f(x) = x, g(x) = cos 2x

True or False? In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The dot product is the only inner product that can be defined in \(R^{n}\). (b) A nonzero vector in an inner product can have a norm of zero. Text Transcription: R^n

True or False? In Exercises 85 and 86, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The norm of the vector \(\mathbf{u}\) is the angle between \(\mathbf{u}\) and the positive x-axis. (b) The angle \(\theta\) between a vector \(\mathbf{v}\) and the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) is obtuse when the scalar a < 0 and acute when a > 0, where \(a \mathbf{v}=\operatorname{proj}_{\mathbf{v}} \mathbf{u}\). Text Transcription: u theta v av = proj_{v} u

Let \(\mathbf{u}=(4,2)\) and \(\mathbf{v}=(2,-2)\) be vectors in \(R^{2}\) with the inner product \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+2 u_{2} v_{2}\). (a) Show that \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. (b) Sketch \(\mathbf{u}\) and \(\mathbf{v}\). Are they orthogonal in the Euclidean sense? Text Transcription: u = (4, 2) v = (2, -2) R^2 langle u, v rangle = u_1 v_1 + 2 u_2 v_2 u v

Proof Prove that \(\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}\) for any vectors \(\mathbf{u}\) and \(\mathbf{v}\) in an inner product space V. Text Transcription: ||u + v||^2 + ||u - v||^2 = 2 ||u ||^2 + 2||v||^2 u v

Proof Prove that the function is an inner product on \(R^{n}\). \(\langle\mathbf{u}, \mathbf{v}\rangle=c_{1} u_{1} v_{1}+c_{2} u_{2} v_{2}+\cdots+c_{n} u_{n} v_{n}, \quad c_{i}>0\) Text Transcription: langle u, v rangle = c_1 u_1 v_1 + c_2 u_2 v_2 + cdots + c_n u_n v_n, c_i > 0

Proof Let \(\mathbf{u}\) and \(\mathbf{v}\) be nonzero vectors in an inner product space V. Prove that \(\mathbf{u}- proj_{\mathbf{v}} \mathbf{u}\) is orthogonal to \(\mathbf{v}\). Text Transcription: u v u - proj_{v} u

Proof Prove Property 2 of Theorem 5.7: If \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w}\) are vectors in an inner product space V, then \(\langle\mathbf{u}+\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{w}\rangle+\langle\mathbf{v}, \mathbf{w}\rangle\). Text Transcription: u v w langle u + v, w rangle = langle u, w rangle + langle v, w rangle

Proof Prove Property 3 of Theorem 5.7: If \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in an inner product space V and c is any real number, then \(\langle\mathbf{u}, c \mathbf{v}\rangle=c\langle\mathbf{u}, \mathbf{v}\rangle\). Text Transcription: u v langle u, cv rangle = c langle u, v rangle

Guided Proof Let W be a subspace of the inner product space V. Prove that the set \(W^{\perp}=\{\mathbf{v} \in V:\langle\mathbf{v}, \mathbf{w}\rangle=0\) for all \(\mathbf{w} \in W\}\) is a subspace of V. Getting Started: To prove that \(W^{\perp}\) is a subspace of V, you must show that \(W^{\perp}\) is nonempty and that the closure conditions for a subspace hold (Theorem 4.5). (i) Find a vector in \(W^{\perp}\) to conclude that it is nonempty. (ii) To show the closure of \(W^{\perp}\) under addition, you need to show that \(\left\langle\mathbf{v}_{1}+\mathbf{v}_{2}, \mathbf{w}\right\rangle=0\) for all \(\mathbf{w} \in W\) and for any \(\mathbf{v}_{1}, \mathbf{v}_{2} \in W^{\perp}\). Use the properties of inner products and the fact that \(\left\langle\mathbf{v}_{1}, \mathbf{w}\right\rangle\) and \(\left\langle\mathbf{v}_{2}, \mathbf{w}\right\rangle\) are both zero to show this. (iii) To show closure under multiplication by a scalar, proceed as in part (ii). Use the properties of inner products and the condition of belonging to \(W^{\perp}\). Text Transcription: W^{perp} = v in V: langle v, w rangle = 0 w in W W^{perp} langle v_1 + v_2, w rangle = 0 v_1, v_2 in W^{perp} langle v_1, w rangle langle v_2, w rangle

Use the result of Exercise 93 to find \(W^{\perp}\) when W is the span of (1, 2, 3) in \(V=R^{3}\). Text Transcription: W^{perp} V = R^3

Guided Proof Let \(\langle\mathbf{u}, \mathbf{v}\rangle\) be the Euclidean inner product on \(R^{n}\). Use the fact that \(\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u}^{T} \mathbf{v}\) to prove that for any n X n matrix A, (a) \(\left\langle A^{T} A \mathbf{u}, \mathbf{v}\right\rangle=\langle\mathbf{u}, A \mathbf{v}\rangle\) and (b) \(\left\langle A^{T} A \mathbf{u}, \mathbf{u}\right\rangle=\|A \mathbf{u}\|^{2}\). Getting Started: To prove (a) and (b), make use of both the properties of transposes (Theorem 2.6) and the properties of the dot product (Theorem 5.3). (i) To prove part (a), make repeated use of the property \(\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u}^{T} \mathbf{v}\) and Property 4 of Theorem 2.6. (ii) To prove part (b), make use of the property \(\langle\mathbf{u}, \mathbf{v}\rangle=\mathbf{u}^{T} \mathbf{v}\), Property 4 of Theorem 2.6, and Property 4 of Theorem 5.3. Text Transcription: langle u, v rangle R^n langle u, v rangle = u^{T} v langle A^{T} Au, v rangle = langle u, Av rangle langle A^{T} Au, u rangle = ||Au||^2 langle u, v rangle = u^{T} v

CAPSTONE (a) Explain how to determine whether a function defines an inner product. (b) Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors in an inner product space V, such that \(\mathbf{v} \neq \mathbf{0}\). Explain how to find the orthogonal projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Text Transcription: u v v neq 0

Finding Inner Product Weights In Exercises 97 - 100, find \(c_{1}\) and \(c_{2}\) for the inner product of \(R^{2}\), \(\langle u, v\rangle=c_{1} u_{1} v_{1}+c_{2} u_{2} v_{2}\) such that the graph represents a unit circle as shown. Text Transcription: c_1 c_2 R^2 langle u, v rangle = c_1 u_1 v_1 + c_2 u_2 v_2

Finding Inner Product Weights In Exercises 97 - 100, find \(c_{1}\) and \(c_{2}\) for the inner product of \(R^{2}\), \(\langle u, v\rangle=c_{1} u_{1} v_{1}+c_{2} u_{2} v_{2}\) such that the graph represents a unit circle as shown. Text Transcription: c_1 c_2 R^2 langle u, v rangle = c_1 u_1 v_1 + c_2 u_2 v_2

Finding Inner Product Weights In Exercises 97 - 100, find \(c_{1}\) and \(c_{2}\) for the inner product of \(R^{2}\), \(\langle u, v\rangle=c_{1} u_{1} v_{1}+c_{2} u_{2} v_{2}\) such that the graph represents a unit circle as shown. Text Transcription: c_1 c_2 R^2 langle u, v rangle = c_1 u_1 v_1 + c_2 u_2 v_2

Finding Inner Product Weights In Exercises 97 - 100, find \(c_{1}\) and \(c_{2}\) for the inner product of \(R^{2}\), \(\langle u, v\rangle=c_{1} u_{1} v_{1}+c_{2} u_{2} v_{2}\) such that the graph represents a unit circle as shown. Text Transcription: c_1 c_2 R^2 langle u, v rangle = c_1 u_1 v_1 + c_2 u_2 v_2

Consider the vectors u = (6, 2, 4) and v = (1, 2, 0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the vector closest to uthat is, show that d(u, projvu) is a minimum.