Finding the Volume of a Parallelepiped In Exercises6972, find the volume V of the

Finding Lengths, Dot Product, and Distance In Exercises 18, find (a) !u!, (b) !v!, (c) u v, and (d) d(u, v). u = (1, 4), v = (2, 1)

Finding Lengths, Dot Product, and Distance In Exercises 18, find (a) !u!, (b) !v!, (c) u v, and (d) d(u, v). u = (1, 2), v = (2, 3)

Finding Lengths, Dot Product, and Distance In Exercises 18, find (a) !u!, (b) !v!, (c) u v, and (d) d(u, v). u = (2, 1, 1), v = (3, 2, 1)

Finding Lengths, Dot Product, and Distance In Exercises 18, find (a) !u!, (b) !v!, (c) u v, and (d) d(u, v). u = (3, 2, 2), v = (1, 3, 5)

Finding Lengths, Dot Product, and Distance In Exercises 18, find (a) !u!, (b) !v!, (c) u v, and (d) d(u, v). u = (1, 2, 0, 1), v = (1, 1, 1, 0)

Finding Lengths, Dot Product, and Distance In Exercises 18, find (a) !u!, (b) !v!, (c) u v, and (d) d(u, v). u = (1, 2, 2, 0), v = (2, 1, 0, 2)

Finding Lengths, Dot Product, and Distance In Exercises 18, find (a) !u!, (b) !v!, (c) u v, and (d) d(u, v). u = (0, 1, 1, 1, 2), v = (0, 1, 2, 1, 1)

Finding Lengths, Dot Product, and Distance In Exercises 18, find (a) !u!, (b) !v!, (c) u v, and (d) d(u, v). u = (1, 1, 0, 1, 1), v = (0, 1, 2, 2, 1)

Finding Length and a Unit Vector In Exercises 912, find !v! and find a unit vector in the direction of v. v = (5, 3, 2)

Finding Length and a Unit Vector In Exercises 912, find !v! and find a unit vector in the direction of v. v = (1, 4, 1)

Finding Length and a Unit Vector In Exercises 912, find !v! and find a unit vector in the direction of v. v = (1, 1, 2)

Finding Length and a Unit Vector In Exercises 912, find !v! and find a unit vector in the direction of v. v = (0, 2, 1)

Consider the vector v = (8, 8, 6). Find u such that (a) u has the same direction as v and one-half its length. (b) u has the direction opposite that of v and one-fourth its length. (c) u has the direction opposite that of v and twice its length

For what values of c is !c(2, 2, 1)! = 3?

Finding the Angle Between Two Vectors In Exercises 1520, find the angle between the two vectors. u = (3, 3), v = (2, 2)

Finding the Angle Between Two Vectors In Exercises 1520, find the angle between the two vectors. u = (1, 1), v = (0, 1)

Finding the Angle Between Two Vectors In Exercises 1520, find the angle between the two vectors. u = (cos 3 4 , sin 3 4 ), v = (cos 2 3 , sin 2 3 )

Finding the Angle Between Two Vectors In Exercises 1520, find the angle between the two vectors. u = (cos 6 , sin 6), v = (cos 5 6 , sin 5 6 )

Finding the Angle Between Two Vectors In Exercises 1520, find the angle between the two vectors. u = (10, 5, 15), v = (2, 1, 3)

Finding the Angle Between Two Vectors In Exercises 1520, find the angle between the two vectors. u = (0, 4, 0, 1), v = (1, 1, 3, 3)

Finding Orthogonal Vectors In Exercises 2124, determine all vectors v that are orthogonal to u. u = (0, 4, 3)

Finding Orthogonal Vectors In Exercises 2124, determine all vectors v that are orthogonal to u. u = (1, 2, 1)

Finding Orthogonal Vectors In Exercises 2124, determine all vectors v that are orthogonal to u. u = (2, 1, 1, 2)

Finding Orthogonal Vectors In Exercises 2124, determine all vectors v that are orthogonal to u. u = (0, 1, 2, 1)

For u = (4, 3 2, 1) and v = ( 1 2, 3, 1), (a) find the inner product represented by u, v = u1v1 + 2u2v2 + 3u3v3, and (b) use this inner product to find the distance between u and v.

For u = (0, 3, 1 3) and v = ( 4 3, 1, 3), (a) find the inner product represented by u, v = 2u1v1 + u2v2 + 2u3v3 and (b) use this inner product to find the distance between u and v.

Verify the triangle inequality and the Cauchy-Schwarz Inequality for u and v from Exercise 25. (Use the inner product given in Exercise 25.)

Verify the triangle inequality and the Cauchy-Schwarz Inequality for u and v from Exercise 26. (Use the inner product given in Exercise 26.)

Calculus In Exercises 29 and 30, (a) find the inner product, (b) determine whether the vectors are orthogonal, and (c) verify the Cauchy-Schwarz Inequality for the vectors. f(x) = x, g(x) = 1 x2 + 1 , f, g = 1 1 f(x)g(x) dx

Calculus In Exercises 29 and 30, (a) find the inner product, (b) determine whether the vectors are orthogonal, and (c) verify the Cauchy-Schwarz Inequality for the vectors. f(x) = x, g(x) = 4x2, f, g = 1 0 f(x)g(x) dx

Finding an Orthogonal Projection In Exercises 3136, find projvu. u = (2, 4), v = (1, 5)

Finding an Orthogonal Projection In Exercises 3136, find projvu. u = (2, 3), v = (0, 4)

Finding an Orthogonal Projection In Exercises 3136, find projvu. u = (2, 5), v = (0, 5)

Finding an Orthogonal Projection In Exercises 3136, find projvu. u = (2, 1), v = (7, 6)

Finding an Orthogonal Projection In Exercises 3136, find projvu. u = (0, 1, 2), v = (3, 2, 4)

Finding an Orthogonal Projection In Exercises 3136, find projvu. u = (1, 3, 1), v = (4, 0, 5)

Applying the Gram-Schmidt Process In Exercises 3740, apply the Gram-Schmidt orthonormalization process to transform the given basis for Rn into an orthonormal basis. Use the Euclidean inner product for Rn and use the vectors in the order in which they are given. B = {(1, 1), (0, 2)}

Applying the Gram-Schmidt Process In Exercises 3740, apply the Gram-Schmidt orthonormalization process to transform the given basis for Rn into an orthonormal basis. Use the Euclidean inner product for Rn and use the vectors in the order in which they are given. B = {(3, 4), (1, 2)}

Applying the Gram-Schmidt Process In Exercises 3740, apply the Gram-Schmidt orthonormalization process to transform the given basis for Rn into an orthonormal basis. Use the Euclidean inner product for Rn and use the vectors in the order in which they are given. B = {(0, 3, 4), (1, 0, 0), (1, 1, 0)}

Applying the Gram-Schmidt Process In Exercises 3740, apply the Gram-Schmidt orthonormalization process to transform the given basis for Rn into an orthonormal basis. Use the Euclidean inner product for Rn and use the vectors in the order in which they are given. B = {(0, 0, 2), (0, 1, 1), (1, 1, 1)}

Let B = {(0, 2, 2), (1, 0, 2)} be a basis for a subspace of R3, and consider x = (1, 4, 2), a vector in the subspace. (a) Write x as a linear combination of the vectors in B. That is, find the coordinates of x relative to B. (b) Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. (c) Write x as a linear combination of the vectors in B. That is, find the coordinates of x relative to B.

Repeat Exercise 41 for B = {(1, 2, 2), (1, 0, 0)} and x = (3, 4, 4).

Calculus In Exercises 4346, let f and g be functions in the vector space C[a, b] with inner product f, g = b a f(x)g(x) dx. Show that f(x) = sin x and g(x) = cos x are orthogonal in C[0, ].

Calculus In Exercises 4346, let f and g be functions in the vector space C[a, b] with inner product f, g = b a f(x)g(x) dx. Show that f(x) = 1 x2 and g(x) = 2x1 x2 are orthogonal in C[1, 1].

Calculus In Exercises 4346, let f and g be functions in the vector space C[a, b] with inner product f, g = b a f(x)g(x) dx. Let f(x) = x and g(x) = x3 be vectors in C[0, 1]. (a) Find f, g. (b) Find !g!. (c) Find d( f, g). (d) Orthonormalize the set B = { f, g}.

Calculus In Exercises 4346, let f and g be functions in the vector space C[a, b] with inner product f, g = b a f(x)g(x) dx. Let f(x) = x + 2 and g(x) = 15x 8 be vectors in C[0, 1]. (a) Find f, g. (b) Find 4f, g. (c) Find ! f !. (d) Orthonormalize the set B = { f, g}.

Find an orthonormal basis for the subspace of Euclidean 3-space below. W = {(x1, x2, x3): x1 + x2 + x3 = 0}

Find an orthonormal basis for the solution space of the homogeneous system of linear equations. x + y z + w = 0 2x y + z + 2w = 0

Proof Prove that if u, v, and w are vectors in Rn, then (u + v) w = u w + v w.

Proof Prove that if u and v are vectors in Rn, then !u + v!2 + !u v!2 = 2!u!2 + 2!v!2.

Proof Prove that if u and v are vectors in an inner product space such that !u! 1 and !v! 1, then u, v 1.

Proof Prove that if u and v are vectors in an inner product space V, then !u!!v! !u v!.

Proof Let V be an m-dimensional subspace of Rn such that m < n. Prove that any vector u in Rn can be uniquely written in the form u = v + w, where v is in V and w is orthogonal to every vector in V

Let V be the two-dimensional subspace of R4 spanned by (0, 1, 0, 1) and (0, 2, 0, 0). Write the vector u = (1, 1, 1, 1) in the form u = v + w, where v is in V and w is orthogonal to every vector in V.

Proof Let {u1, u2, . . . , um} be an orthonormal subset of Rn, and let v be any vector in Rn. Prove that !v!2 m i=1 (v ui )2. (This inequality is called Bessels Inequality.)

Proof Let {x1, x2, . . . , xn} be a set of real numbers. Use the Cauchy-Schwarz Inequality to prove that (x1 + x2 + . . . + xn)2 n(x2 1 + x2 2 + . . . + x2 n).

Proof Let u and v be vectors in an inner product space V. Prove that !u + v! = !u v! if and only if u and v are orthogonal.

Writing Let {u1, u2, . . . , un} be a dependent set of vectors in an inner product space V. Describe the result of applying the Gram-Schmidt orthonormalization process to this set.

Find the orthogonal complement S of the subspace S of R3 spanned by the two column vectors of the matrix A = [ 1 2 0 2 1 1 ]

Find the projection of the vector v = [1 0 2]T onto the subspace S = span{[ 0 1 1 ] , [ 0 1 1 ]}.

Find bases for the four fundamental subspaces of the matrix A = [ 0 0 1 1 3 0 0 0 1 ] .

Find the least squares regression line for the set of data points {(2, 2), (1, 1), (0, 1), (1, 3)}. Graph the points and the line on the same set of axes.

Revenue The table shows the revenues y (in billions of dollars) for Google, Incorporated from 2006 through 2013. Find the least squares regression cubic polynomial for the data. Then use the model to predict the revenue in 2018. Let t represent the year, with t = 6 corresponding to 2006. (Source: Google, Incorporated) Year 2006 2007 2008 2009 Revenue, y 10.6 16.6 21.8 23.7 Year 2010 2011 2012 2013 Revenue, y 29.3 37.9 50.2 59.8

Petroleum Production The table shows the North American petroleum productions y (in millions of barrels per day) from 2006 through 2013. Find the least squares regression linear and quadratic polynomials for the data. Then use the model to predict the petroleum production in 2018. Let t represent the year, with t = 6 corresponding to 2006. Which model appears to be more accurate for predicting future petroleum productions? Explain. (Source: U.S. Energy Information Administration) Year 2006 2007 2008 2009 Petroleum Production, y 15.3 15.4 15.1 15.4 Year 2010 2011 2012 2013 Petroleum Production, y 16.1 16.7 17.9 19.3

Finding the Cross Product In Exercises 6568, find u v and show that it is orthogonal to both u and v. u = (1, 1, 0), v = (0, 3, 0)

Finding the Cross Product In Exercises 6568, find u v and show that it is orthogonal to both u and v. u = (1, 1, 1), v = (0, 1, 1)

Finding the Cross Product In Exercises 6568, find u v and show that it is orthogonal to both u and v. u = j + 6k, v = i 2j + k

Finding the Cross Product In Exercises 6568, find u v and show that it is orthogonal to both u and v. u = 2i k, v = i + j k

Finding the Volume of a Parallelepiped In Exercises 6972, find the volume V of the parallelepiped that has u, v, and w as adjacent edges using the formula V = u (v w). u = (1, 0, 0) v = (0, 0, 1) w = (0, 1, 0)

Finding the Volume of a Parallelepiped In Exercises 6972, find the volume V of the parallelepiped that has u, v, and w as adjacent edges using the formula V = u (v w). u = (1, 2, 1) v = (1, 1, 0) w = (3, 4, 1)

Finding the Volume of a Parallelepiped In Exercises 6972, find the volume V of the parallelepiped that has u, v, and w as adjacent edges using the formula V = u (v w). u = 2i + j v = 3i 2j + k w = 2i 3j 2k

Finding the Volume of a Parallelepiped In Exercises 6972, find the volume V of the parallelepiped that has u, v, and w as adjacent edges using the formula V = u (v w). u = i + j + 3k v = 3j + 3k w = 3i + 3k

Find the area of the parallelogram that has u = (1, 3, 0) and v = (1, 0, 2) as adjacent sides.

Proof Prove that !u v! = !u! !v! if and only if u and v are orthogonal.

Finding a Least Squares Approximation In Exercises 7578, (a) find the least squares approximation g(x) = a0 + a1x of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. f(x) = x3, 1 x 1

Finding a Least Squares Approximation In Exercises 7578, (a) find the least squares approximation g(x) = a0 + a1x of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. f(x) = x3, 0 x 2

Finding a Least Squares Approximation In Exercises 7578, (a) find the least squares approximation g(x) = a0 + a1x of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. f(x) = sin 2x, 0 x :2

Finding a Least Squares Approximation In Exercises 7578, (a) find the least squares approximation g(x) = a0 + a1x of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. f(x) = sin x cos x, 0 x

Finding a Least Squares Approximation In Exercises 79 and 80, (a) find the least squares approximation g(x) = a0 + a1x + a2x2 of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. f(x) = x, 0 x 1

Finding a Least Squares Approximation In Exercises 79 and 80, (a) find the least squares approximation g(x) = a0 + a1x + a2x2 of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. f(x) = 1 x , 1 x 2

Finding a Fourier Approximation In Exercises 81 and 82, find the Fourier approximation with the specified order of the function on the interval [, ]. f(x) = x2, first order

Finding a Fourier Approximation In Exercises 81 and 82, find the Fourier approximation with the specified order of the function on the interval [, ]. f(x) = x, second order

True or False? In Exercises 83 and 84, 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 cross product of two nonzero vectors in R3 yields a vector orthogonal to the two vectors that produced it. (b) The cross product of two nonzero vectors in R3 is commutative. (c) The least squares approximation of a function f is the function g (in the subspace W) closest to f in terms of the inner product f, g.

True or False? In Exercises 83 and 84, 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 vectors u v and v u in R3 have equal lengths but opposite directions. (b) If u and v are two nonzero vectors in R3, then u and v are parallel if and only if u v = 0. (c) A special type of least squares approximation, the Fourier approximation, is spanned by the basis S = {1, cos x, cos 2x, . . . , cos nx, sin x, sin 2x, . . . , sin nx}

In Exercises 1 - 8, find (a) \(\|\mathbf{u}\|\), (b) \(\|\mathbf{v}\|\), (c) \(\mathbf{u} \cdot \mathbf{v}\), and (d) \(d(\mathbf{u}, \mathbf{v})\). \(\mathbf{u}=(1,4), \quad \mathbf{v}=(2,1)\) Text Transcription: ||u|| ||v|| u cdot v d(u, v) u = (1, 4), v = (2,1)

In Exercises 1 - 8, find (a) \(\|\mathbf{u}\|\), (b) \(\|\mathbf{v}\|\), (c) \(\mathbf{u} \cdot \mathbf{v}\), and (d) \(d(\mathbf{u}, \mathbf{v})\). \(\mathbf{u}=(-1,2), \quad \mathbf{v}=(2,3)\) Text Transcription: ||u|| ||v|| u cdot v d(u, v) u = (-1, 2), v = (2, 3)

In Exercises 1 - 8, find (a) \(\|\mathbf{u}\|\), (b) \(\|\mathbf{v}\|\), (c) \(\mathbf{u} \cdot \mathbf{v}\), and (d) \(d(\mathbf{u}, \mathbf{v})\). \(\mathbf{u}=(2,1,1), \quad \mathbf{v}=(3,2,-1)\) Text Transcription: ||u|| ||v|| u cdot v d(u, v) u = (2,1,1), v = (3, 2, -1)

In Exercises 1 - 8, find (a) \(\|\mathbf{u}\|\), (b) \(\|\mathbf{v}\|\), (c) \(\mathbf{u} \cdot \mathbf{v}\), and (d) \(d(\mathbf{u}, \mathbf{v})\). \(\mathbf{u}=(-3,2,-2), \quad \mathbf{v}=(1,3,5)\) Text Transcription: ||u|| ||v|| u cdot v d(u, v) u = (-3, 2, -2), v = (1, 3, 5)

Finding Lengths, Dot Product, and Distance In Exercises 18, find (a) !u!, (b) !v!, (c) u v, and (d) d(u, v). u = (1, 2, 0, 1), v = (1, 1, 1, 0)

In Exercises 1 - 8, find (a) \(\|\mathbf{u}\|\), (b) \(\|\mathbf{v}\|\), (c) \(\mathbf{u} \cdot \mathbf{v}\), and (d) \(d(\mathbf{u}, \mathbf{v})\). \(\mathbf{u}=(1,-2,2,0), \quad \mathbf{v}=(2,-1,0,2)\) Text Transcription: ||u|| ||v|| u cdot v d(u, v) u = (1, -2, 2, 0), v = (2, -1, 0, 2)

In Exercises 1 - 8, find (a) \(\|\mathbf{u}\|\), (b) \(\|\mathbf{v}\|\), (c) \(\mathbf{u} \cdot \mathbf{v}\), and (d) \(d(\mathbf{u}, \mathbf{v})\). \(\mathbf{u}=(0,1,-1,1,2), \quad \mathbf{v}=(0,1,-2,1,1)\) Text Transcription: ||u|| ||v|| u cdot v d(u, v) u = (0, 1, -1, 1, 2), v = (0, 1, -2, 1, 1)

In Exercises 1 - 8, find (a) \(\|\mathbf{u}\|\), (b) \(\|\mathbf{v}\|\), (c) \(\mathbf{u} \cdot \mathbf{v}\), and (d) \(d(\mathbf{u}, \mathbf{v})\). \(\mathbf{u}=(1,-1,0,1,1), \quad \mathbf{v}=(0,1,-2,2,1)\) Text Transcription: ||u|| ||v|| u cdot v d(u, v) u = (1, -1, 0, 1, 1), v = (0, 1, -2, 2, 1)

Finding Length and a Unit Vector In Exercises 9-12, find \(\|\mathbf{v}\|\) and find a unit vector in the direction of v. \(\mathbf{v}=(5,3,-2)\) Text Transcription: ||v|| v = (5, 3, -2)

Finding Length and a Unit Vector In Exercises 9-12, find \(\|\mathbf{v}\|\) and find a unit vector in the direction of v. \(\mathbf{v}=(-1,-4,1)\) Text Transcription: ||v|| v = (-1,-4,1)\)

Finding Length and a Unit Vector In Exercises 9-12, find \(\|\mathbf{v}\|\) and find a unit vector in the direction of v. \(\mathbf{v}=(-1,1,2)\) Text Transcription: ||v|| v = (-1,1,2)

Finding Length and a Unit Vector In Exercises 9-12, find \(\|\mathbf{v}\|\) and find a unit vector in the direction of v. \(\mathbf{v}=(0,2,-1)\) Text Transcription: ||v|| v = (0, 2, -1)

Consider the vector \(\mathbf{v}=(8,8,6)\). Find \(\mathbf{u}\) such that (a) \(\mathbf{u}\) has the same direction as \(\mathbf{v}\) and one-half its length. (b) \(\mathbf{u}\) has the direction opposite that of \(\mathbf{v}\) and one-fourth its length. (c) \(\mathbf{u}\) has the direction opposite that of \(\mathbf{v}\) and twice its length. Text Transcription: v = (8, 8, 6) u v

For what values of c is |c(2, 2, -1)| = 3?

Finding the Angle Between Two Vectors In Exercises 15 - 20, find the angle \(\theta\) between the two vectors. \(\mathbf{u}=(3,3), \quad \mathbf{v}=(-2,2)\) Text Transcription: u = (3, 3), v = (-2, 2)

Finding the Angle Between Two Vectors In Exercises 15 - 20, find the angle \(\theta\) between the two vectors. \(\mathbf{u}=(1,-1), \quad \mathbf{v}=(0,1)\) Text Transcription: theta u = (1, -1), v = (0, 1)

Finding the Angle Between Two Vectors In Exercises 15 - 20, find the angle \(\theta\) between the two vectors. \(\mathbf{u}=\left(\cos \frac{3 \pi}{4}, \sin \frac{3 \pi}{4}\right), \quad \mathbf{v}=\left(\cos \frac{2 \pi}{3}, \sin \frac{2 \pi}{3}\right)\) Text Transcription: theta u = (cos 3 pi / 4, sin 3 pi / 4), v = (cos 2 pi / 3, sin 2 pi / 3)

Finding the Angle Between Two Vectors In Exercises 15 - 20, find the angle \(\theta\) between the two vectors. \(\mathbu = \left(\cos \frac{\pi}{6}, \sin \frac{\pi}{6}\right), \quad \mathbf{v}=\left(\cos \frac{5 \pi}{6}, \sin \frac{5 \pi}{6}\right)\) Text Transcription: theta u = (cos pi / 6, sin pi / 6), v = (cos 5 pi / 6, sin 5 pi / 6)

Finding the Angle Between Two Vectors In Exercises 15 - 20, find the angle \(\theta\) between the two vectors. \(\mathbf{u}=(10,-5,15), \quad \mathbf{v}=(-2,1,-3)\) Text Transcription: theta u = (10, -5, 15), v = (-2, 1, -3)

Finding the Angle Between Two Vectors In Exercises 15 - 20, find the angle \(\theta\) between the two vectors. \(\mathbf{u}=(0,4,0,-1), \quad \mathbf{v}=(1,1,3,-3)\) Text Transcription: theta u = (0, 4, 0, -1), v = (1, 1, 3, -3)

Finding Orthogonal Vectors In Exercises 21 - 24, determine all vectors v that are orthogonal to u. \(\mathbf{u}=(0,-4,3)\) Text Transcription: u = (0, -4, 3)

Finding Orthogonal Vectors In Exercises 21 - 24, determine all vectors v that are orthogonal to u. \(\mathbf{u}=(1,-2,1)\) Text Transcription: u = (1, -2, 1)

Finding Orthogonal Vectors In Exercises 21 - 24, determine all vectors v that are orthogonal to u. \(\mathbf{u}=(2,-1,1,2)\) Text Transcription: u = (2, -1, 1, 2)

Finding Orthogonal Vectors In Exercises 21 - 24, determine all vectors v that are orthogonal to u. \(\mathbf{u}=(0,1,2,-1)\) Text Transcription: u = (0, 1, 2, -1)

For \(\mathbf{u}=\left(4,-\frac{3}{2},-1\right)\) and \(\mathbf{v}=\left(\frac{1}{2}, 3,1\right)\), (a) find the inner product represented by \(\langle\mathbf{u}, \mathbf{v}\rangle=u_{1} v_{1}+2 u_{2} v_{2}+3 u_{3} v_{3}\), and (b) use this inner product to find the distance between \(\mathbf{u}\) and \(\mathbf{v}\). Text Transcription: u = (4, -3 / 2, -1) v = (1 / 2, 3,1) langle u}, v rangle = u_1 v_1 + 2u_2 v_2 + 3u_3 v_3 u v

For \(\mathbf{u}=\left(0,3, \frac{1}{3}\right)\) and \(\mathbf{v}=\left(\frac{4}{3}, 1,-3\right)\), (a) find the inner product represented by \(\langle\mathbf{u}, \mathbf{v}\rangle=2 u_{1} v_{1}+u_{2} v_{2}+2 u_{3} v_{3}\) and (b) use this inner product to find the distance between \(\mathbf{u}\) and \(\mathbf{v}\). Text Transcription: u = (0, 3, 1 / 3) v = (4 / 3, 1, -3) langle u, v rangle = 2u_1 v_1 + u_2 v_2 + 2u_3 v_3 u v

Verify the triangle inequality and the Cauchy-Schwarz Inequality for \(\mathbf{u}\) and \(\mathbf{v}\) from Exercise 25 . (Use the inner product given in Exercise 25.) Text Transcription: u v

Verify the triangle inequality and the Cauchy-Schwarz Inequality for \(\mathbf{u}\) and \(\mathbf{v}\) from Exercise 26. (Use the inner product given in Exercise 26.) Text Transcription: u v

Calculus In Exercises 29 and 30, (a) find the inner product, (b) determine whether the vectors are orthogonal, and (c) verify the Cauchy-Schwarz Inequality for the vectors. \(f(x)=x, g(x)=\frac{1}{x^{2}+1},\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\) Text Transcription: f(x) = x, g(x) = 1 / x^2 + 1, langle f, g rangle = int_-1^{1} f(x) g(x) dx

Calculus In Exercises 29 and 30, (a) find the inner product, (b) determine whether the vectors are orthogonal, and (c) verify the Cauchy-Schwarz Inequality for the vectors. \(f(x)=x, g(x)=4 x^{2},\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x\) Text Transcription: f(x) = x, g(x) = 4 x^{2},langle f, g rangle = int_{0}^{1} f(x) g(x) dx

Finding an Orthogonal Projection In Exercises 31-36, find \(\mathbf{p r o j}_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u}=(2,4), \quad \mathbf{v}=(1,-5)\) Text Transcription: proj_{v} u u = (2, 4), v = (1, -5)

Finding an Orthogonal Projection In Exercises 31-36, find \(\mathbf{p r o j}_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u}=(2,3), \quad \mathbf{v}=(0,4)\) Text Transcription: proj_{v} u u =(2,3), v = (0,4)

Finding an Orthogonal Projection In Exercises 31-36, find \(\mathbf{p r o j}_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u}=(2,5), \quad \mathbf{v}=(0,5)\) Text Transcription: proj_{v} u u = (2, 5), v = (0, 5)

Finding an Orthogonal Projection In Exercises 31-36, find \(\mathbf{p r o j}_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u}=(2,-1), \quad \mathbf{v}=(7,6)\) Text Transcription: proj_{v} u u = (2, -1), v = (7, 6)

Finding an Orthogonal Projection In Exercises 31-36, find \(\mathbf{p r o j}_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u}=(0,-1,2), \quad \mathbf{v}=(3,2,4)\) Text Transcription: proj_{v} u u = (0, -1, 2), v = (3, 2, 4)

Finding an Orthogonal Projection In Exercises 31-36, find \(\mathbf{p r o j}_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u}=(-1,3,1), \quad \mathbf{v}=(4,0,5)\) Text Transcription: proj_{v} u u = (-1, 3, 1), v = (4, 0, 5)

Applying the Gram-Schmidt Process In Exercises 37 - 40, apply the Gram-Schmidt orthonormalization process to transform the given basis for \(R^{n}\) into an orthonormal basis. Use the Euclidean inner product for \(R^{n}\) and use the vectors in the order in which they are given. B = {(1,1), (0,2)} Text Transcription: R^n

Applying the Gram-Schmidt Process In Exercises 37 - 40, apply the Gram-Schmidt orthonormalization process to transform the given basis for \(R^{n}\) into an orthonormal basis. Use the Euclidean inner product for \(R^{n}\) and use the vectors in the order in which they are given. B = {(3,4), (1,2)} Text Transcription: R^n

Applying the Gram-Schmidt Process In Exercises 37 - 40, apply the Gram-Schmidt orthonormalization process to transform the given basis for \(R^{n}\) into an orthonormal basis. Use the Euclidean inner product for \(R^{n}\) and use the vectors in the order in which they are given. B = {(0, 3, 4), (1, 0, 0), (1, 1, 0)} Text Transcription: R^n

Applying the Gram-Schmidt Process In Exercises 37 - 40, apply the Gram-Schmidt orthonormalization process to transform the given basis for \(R^{n}\) into an orthonormal basis. Use the Euclidean inner product for \(R^{n}\) and use the vectors in the order in which they are given. B = {(0, 0, 2), (0, 1, 1), (1, 1, 1)} Text Transcription: R^n

Let B ={(0,2,-2),(1,0,-2)} be a basis for a subspace of \(R^{3}\), and consider \(\mathbf{x}=(-1,4,-2)\), a vector in the subspace. (a) Write \(\mathbf{x}\) as a linear combination of the vectors in B. That is, find the coordinates of \(\mathbf{x}\) relative to B. (b) Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B’. (c) Write \(\mathbf{x}\) as a linear combination of the vectors in B’. That is, find the coordinates of \(\mathbf{x}\) relative to B’. Text Transcription: R^3 x = (-1, 4, -2) x

Repeat Exercise 41 for B = {(-1, 2, 2), (1, 0, 0)} and \(\mathbf{x}=(-3,4,4)\). Text Transcription: x = (-3, 4, 4)

Calculus In Exercises 43 - 46, let f and g be functions in the vector space C [a, b] with inner product \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) dx\). Show that f(x) = sin x and g(x) = cos x are orthogonal in \(C[0, \pi]\). Text Transcription: langle f, g rangle = int_{a}^{b} f(x) g(x) dx. C [0, pi]

Calculus In Exercises 43 - 46, let f and g be functions in the vector space C [a, b] with inner product \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) dx\). Show that \(f(x)=\sqrt{1-x^{2}}\) and \(g(x)=2 x \sqrt{1-x^{2}}\) are orthogonal in C[-1, 1]. Text Transcription: langle f, g rangle = int_{a}^{b} f(x) g(x) dx f(x) = sqrt{1 - x^2} g(x) = 2x sqrt {1 - x^2}

Calculus In Exercises 43 - 46, let f and g be functions in the vector space C [a, b] with inner product \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) dx\). Let f(x) = x and \(g(x)=x^{3}\) be vectors in C[0, 1]. (a) Find \(\langle f, g\rangle\). (b) Find |g| (c) Find d(f, g) (d) Orthonormalize the set B = {f, g} Text Transcription: langle f, g rangle = int_{a}^{b} f(x) g(x) dx g(x) = x^3 langle f, g rangle

Calculus In Exercises 43 - 46, let f and g be functions in the vector space C [a, b] with inner product \(\langle f, g\rangle=\int_{a}^{b} f(x) g(x) dx\). Let f(x) = x + 2 and g(x) = 15x - 8 be vectors in C[0, 1]. (a) Find \(\langle f, g\rangle\). (b) Find \(\langle-4 f, g\rangle\). (c) Find |f|. (d) Orthonormalize the set B = {f, g}. Text Transcription: langle f, g rangle = int_{a}^{b} f(x) g(x) dx langle f, g rangle langle - 4f, g rangle

Find an orthonormal basis for the subspace of Euclidean 3 - space below. \(W=\left\{\left(x_{1}, x_{2}, x_{3}\right): x_{1}+x_{2}+x_{3}=0\right\}\) Text Transcription: W = {(x_1, x_2, x_3): x_1 + x_2 + x_3 = 0}

Find an orthonormal basis for the solution space of the homogeneous system of linear equations. x + y - z + w = 0 2x - y + z + 2w = 0

Proof Prove that if \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w}\) are vectors in \(R^{n}\), then \((\mathbf{u}+\mathbf{v}) \cdot \mathbf{w}=\mathbf{u} \cdot \mathbf{w}+\mathbf{v} \cdot \mathbf{w}\). Text Transcription: u, v w R^n u + v cdot w = u cdot w + v cdot w

Proof Prove that if \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in \(R^{n}\), then \(\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}\). Text Transcription: u v R^n |u + v|^2 + |u - v|^2 = 2|u|^2 + 2|v|^2

Proof Prove that if \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in an inner product space such that \(\|\mathbf{u}\| \leq 1\) and \(\|\mathbf{v}\| \leq 1\), then \(|\langle\mathbf{u}, \mathbf{v}\rangle| \leq 1\). Text Transcription: u v |u| leq 1 |v| leq 1 |langle u, v rangle| leq 1

Proof Prove that if \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in an inner product space V, then \(|\|\mathbf{u}\|-\|\mathbf{v}\|| \leq\|\mathbf{u} \pm \mathbf{v}\|\) . Text Transcription: u v ||u}| - |v|| leq |u pm v|

Proof Let V be an m-dimensional subspace of \(R^{n}\) such that m < n. Prove that any vector \(\mathbf{u}\) in \(R^{n}\) can be uniquely written in the form \(\mathbf{u}=\mathbf{v}+\mathbf{w}\), where \(\mathbf{v}\) is in V and \(\mathbf{w}\) is orthogonal to every vector in V. Text Transcription: R^n u u = v + w v w

Let V be the two-dimensional subspace of \(R^{4}\) spanned by (0, 1, 0, 1) and (0, 2, 0, 0). Write the vector \(\mathbf{u}=(1,1,1,1)\) in the form \(\mathbf{u}=\mathbf{v}+\mathbf{w}\), where \(\mathbf{v}\) is in V and \(\mathbf{w}\) is orthogonal to every vector in V. Text Transcription: R^4 u = (1, 1, 1, 1) u = v + w v w

Proof Let \(\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{m}\right\}\) be an orthonormal subset of \(R^{n}\), and let \(\mathbf{v}\) be any vector in \(R^{n}\). Prove that \(\|\mathbf{v}\|^{2} \geq \sum_{i=1}^{m}\left(\mathbf{v} \cdot \mathbf{u}_{i}\right)^{2}\) . (This inequality is called Bessel's Inequality.) Text Transcription: {u_1, u_2, ldots, u_m} R^n v |v|^2 geq sum_{i = 1}^{m} (v cdot u_i)^2

Proof Let \(\left\{x_{1}, x_{2}, \ldots , x_{n}\right\}\) be a set of real numbers. Use the Cauchy-Schwarz Inequality to prove that \(\left(x_{1}+x_{2}+\cdots+x_{n}\right)^{2} \leq n\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right)\). Text Transcription: {x_1, x_2, ..., x_{n}\right\}\ (x_1 + x_2 + cdots + x_n)^2 leq n(x_1^2 + x_2^2 + cdots + x_n^2)

Proof Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors in an inner product space V. Prove that \(\|\mathbf{u}+\mathbf{v}\|=\|\mathbf{u}-\mathbf{v}\|\) if and only if \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. Text Transcription: u v |u + v| = |u - v|

Writing Let \(\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, . . . \mathbf{u}_{n}\right\}\) be a dependent set of vectors in an inner product space V. Describe the result of applying the Gram-Schmidt orthonormalization process to this set. Text Transcription: {u_1, u_2}, . . . u_n}

Find the orthogonal complement \(S^{\perp}\) of the subspace S of \(R^{3}\) spanned by the two-column vectors of the matrix \(A=\left[\begin{array}{rr} 1 2 \\ 2 1 \\ 0 -1 \end{array}\right]\). Text Transcription: A = [1 2 \\ 2 1 \\ 0 -1]

Find the projection of the vector \(\mathbf{v}=\left[\begin{array}{lll}1 0 -2\end{array}\right]^{T}\) onto the subspace \(S=\operatorname{span}\left\{\left[\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right],\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right]\right\}\) Text Transcription: v = [1 0 -2]^T S = span {[ 0 \\ -1 \\ 1], [0 \\ 1 \\ 1 ]}

Find bases for the four fundamental subspaces of the matrix \(A=\left[\begin{array}{rrr} 0 1 0 \\ 0 -3 0 \\ 1 0 1 \end{array}\right]\). Text Transcription: A = [0 1 0 \\ 0 -3 0 \\ 1 0 1]

Find the least-squares regression line for the set of data points {(-2, 2), (-1, 1), (0, 1), (1, 3)} Graph the points and the line on the same set of axes.

Revenue The table shows the revenues y (in billions of dollars) for Google, Incorporated from 2006 through 2013. Find the least-squares regression cubic polynomial for the data. Then use the model to predict the revenue in 2018. Let t represent the year, with t = 6 corresponding to 2006. (Source: Google, Incorporated)

Petroleum Production The table shows the North American petroleum production y (in millions of barrels per day) from 2006 through 2013. Find the least-squares regression linear and quadratic polynomials for the data. Then use the model to predict the petroleum production in 2018. Let t represent the year, with t = 6 corresponding to 2006. Which model appears to be more accurate for predicting future petroleum productions? Explain. (Source: U.S. Energy Information Administration).

Finding the Cross Product In Exercises 65 - 68, find u X v and show that it is orthogonal to both u and v. \(\mathbf{u}=(1,1,0), \quad \mathbf{v}=(0,3,0)\) Text Transcription: u = (1, 1, 0), v = (0, 3, 0)

Finding the Cross Product In Exercises 65 - 68, find u X v and show that it is orthogonal to both u and v. \(\mathbf{u}=(1,-1,1), \quad \mathbf{v}=(0,1,1)\) Text Transcription: u = (1, -1, 1), v = (0, 1, 1)

Finding the Cross Product In Exercises 65 - 68, find u X v and show that it is orthogonal to both u and v. \(\mathbf{u}=\mathbf{j}+6 \mathbf{k}, \quad \mathbf{v}=\mathbf{i}-2 \mathbf{j}+\mathbf{k}\) Text Transcription: u = j + 6k, v = i - 2j + k

Finding the Cross Product In Exercises 65 - 68, find u X v and show that it is orthogonal to both u and v. \(\mathbf{u}=2 \mathbf{i}-\mathbf{k}, \quad \mathbf{v}=\mathbf{i}+\mathbf{j}-\mathbf{k}\) Text Transcription: u = 2i - k, v = i + j - k

Finding the Volume of a Parallelepiped In Exercises 69 - 72, find the volume V of the parallelepiped that has u, v, and w as adjacent edges using the formula \(\boldsymbol{V}=|\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})|\) \(\mathbf{u}=(1,0,0)\) \(\mathbf{v}=(0,0,1)\) \(\mathbf{w}=(0,1,0)\) Text Transcription: V = |u cdot (v X w)| u = (1, 0, 0) v = (0, 0, 1) w = (0, 1, 0)

Finding the Volume of a Parallelepiped In Exercises 69 - 72, find the volume V of the parallelepiped that has u, v, and w as adjacent edges using the formula \(\boldsymbol{V}=|\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})|\) \(\mathbf{u}=(1,2,1)\) \(\mathbf{v}=(-1,-1,0)\) \(\mathbf{w}=(3,4,-1)\) Text Transcription: V = |u cdot (v X w)| u = (1, 2, 1) v = (-1, -1, 0) w = (3, 4, -1)

Finding the Volume of a Parallelepiped In Exercises 69 - 72, find the volume V of the parallelepiped that has u, v, and w as adjacent edges using the formula \(\boldsymbol{V}=|\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})|\) \(\mathbf{u}=-2 \mathbf{i}+\mathbf{j}\) \(\mathbf{v}=3 \mathbf{i}-2 \mathbf{j}+\mathbf{k}\) \(\mathbf{w}=2 \mathbf{i}-3 \mathbf{j}-2 \mathbf{k}\) Text Transcription: V = |u cdot (v X w)| u = -2i + j v = 3i - 2j + k w = 2i - 3j - 2k

Finding the Volume of a Parallelepiped In Exercises 69 - 72, find the volume V of the parallelepiped that has u, v, and w as adjacent edges using the formula \(\boldsymbol{V}=|\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})|\) \(\mathbf{u}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}\) \(\mathbf{v}=3 \mathbf{j}+3 \mathbf{k}\) \(\mathbf{w}=3 \mathbf{i}+3 \mathbf{k}\) Text Transcription: V = |u cdot (v X w)| u = i + j + 3k v = 3j + 3k w = 3i + 3k

Find the area of the parallelogram that has \(\mathbf{u}=(1,3,0)\) and \(\mathbf{v}=(-1,0,2)\) as adjacent sides. Text Transcription: u = (1, 3, 0) v = (-1, 0, 2)

Proof Prove that \(\||\mathbf{u} \times \mathbf{v}\||=\||\mathbf{u}\||\||\mathbf{v}\||\) if and only if \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. Text Transcription: ||u X v|| = ||u|| ||v|| u v

Finding a Least Squares Approximation In Exercises 75 - 78, (a) find the least-squares approximation \(g(x)=a_{0}+a_{1} x\) of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. \(f(x)=x^{3},-1 \leq x \leq 1\) Text Transcription: g(x) = a_0 + a_1 x f(x) = x^3, -1 leq x leq 1

Finding a Least Squares Approximation In Exercises 75 - 78, (a) find the least-squares approximation \(g(x)=a_{0}+a_{1} x\) of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. \(f(x)=x^{3}, 0 \leq x \leq 2\) Text Transcription: g(x) = a_0 + a_1 x f(x) = x^3, 0 leq x leq 2

Finding a Least Squares Approximation In Exercises 75 - 78, (a) find the least-squares approximation \(g(x)=a_{0}+a_{1} x\) of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. \(f(x)=\sin 2 x, 0 \leq x \leq \pi / 2\) Text Transcription: g(x) = a_0 + a_1 x f(x) = sin 2x, 0 leq x leq pi/2

Finding a Least Squares Approximation In Exercises 75 - 78, (a) find the least-squares approximation \(g(x)=a_{0}+a_{1} x\) of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. \(f(x)=\sin x \cos x, 0 \leq x \leq \pi\) Text Transcription: g(x) = a_0 + a_1 x f(x) = sin x cos x, 0 leq x leq pi

Finding a Least Squares Approximation In Exercises 79 and 80, (a) find the least-squares approximation \(g(x)=a_{0}+a_{1} x+a_{2} x^{2}\) of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. \(f(x)=\sqrt{x}, 0 \leq x \leq 1\) Text Transcription: g(x) = a_0 + a_1 x + a_2 x^2 f(x) = sqrtx, 0 leq x leq 1

Finding a Least Squares Approximation In Exercises 79 and 80, (a) find the least-squares approximation \(g(x)=a_{0}+a_{1} x+a_{2} x^{2}\) of the function f, and (b) use a graphing utility to graph f and g in the same viewing window. \(f(x)=\frac{1}{x}, 1 \leq x \leq 2\) Text Transcription: g(x) = a_0 + a_1 x + a_2 x^2 f(x) = 1 / x, 1 leq x leq 2

Finding a Fourier Approximation In Exercises 81 and 82, find the Fourier approximation with the specified order of the function on the interval \([-\pi, \pi]\). \(f(x)=x^{2}\), first order Text Transcription: [-pi, pi] f(x) = x^2

Finding a Fourier Approximation In Exercises 81 and 82, find the Fourier approximation with the specified order of the function on the interval \([-\pi, \pi]\). f(x) = x, second order Text Transcription: [-pi, pi]

(a) The cross product of two nonzero vectors in \(R^{3}\) yields a vector orthogonal to the two vectors that produced it. (b) The cross product of two nonzero vectors in \(R^{3}\) is commutative. (c) The least-squares approximation of a function f is the function g (in the subspace W) closest to f in terms of the inner product \(\langle f, g\rangle\). Text Transcription: R^3 langle f, g rangle

(a) The vectors \(\mathbf{u} \times \mathbf{v}\) and \(\mathbf{v} \times \mathbf{u}\) in \(R^{3}\) have equal lengths but opposite directions. (b) If \(\mathbf{u}\) and \(\mathbf{v}\) are two nonzero vectors in \(R^{3}\), then \(\mathbf{u}\) and \(\mathbf{v}\) are parallel if and only if \(\mathbf{u} \times \mathbf{v}=\mathbf{0}\). (c) A special type of least squares approximation, the Fourier approximation, is spanned by the basis \(S=\{1, \cos x, \cos 2 x, \ldots, \cos n x, \sin x, \sin 2 x, . ., \sin n x\}\). Text Transcription: u X v v X u R^3 u v u X v = 0 S = {1, cos x, cos 2x, ..., cos nx, sin x, sin 2x, . ., sin nx}