Answer: For Exercises 2330, compute the indicated projection with respectto the given

For Exercises 18, compute the indicated inner product. u, v for u = 1 2 1  , v = 3 4 2  , and the inner product given in Example 2.

For Exercises 18, compute the indicated inner product. u, v for u = 2 5 2  , v = 1 3 0  , and the inner product given in Example 2 with weights t1 = 3, t2 = 1, and t3 = 4.

For Exercises 18, compute the indicated inner product. p, qfor p(x) = 3x+2, q(x) = x+1, and the inner product given in Example 3 with x0 = 1, x1 = 0, and x2 = 2.

For Exercises 18, compute the indicated inner product. . p, qfor p(x) = x2+1, q(x) = 2x 3, and the inner product given in Example 3 with x0 = 1, x1 = 1, x2 = 2, and x3 = 5.

For Exercises 18, compute the indicated inner product. f, g for f (x) = x + 3, g (x) = x2, and the inner product given in Example 5.

For Exercises 18, compute the indicated inner product. f, g for f (x) = x, g (x) = e x , and the inner product given in Example 5.

For Exercises 18, compute the indicated inner product. A, B = tr(AT B) for A =  2 1 3 4 , B =  5 2 3 2 

For Exercises 18, compute the indicated inner product. A, B = tr(AT B) for A = 7 -3 2 6 , B = 4 2 0 5

Suppose that u =  1 0 1  and v = 2 1 2  are orthogonal with respect to the inner product given in Example 2 with weights t1 = 3, t2 = 1, and t3 = a. What are the possible value(s) of a?

Suppose that u = 3 2 2  and v =  5 1 2  are orthogonal with respect to the inner product given in Example 2 with weights t1 = 1, t2 = b, and t3 = 2. What are the possible value(s) of b?

Suppose that p(x) = x + 2, q(x) = 3x + 1 are orthogonal with respect to the inner product given in Example 3 with x0 = 1, x1 = a, and x2 = 2. What are the possible value(s) of a?

Suppose that p(x) = x23x1, q(x) = x+2 are orthogonal with respect to the inner product given in Example 3 with x0 = 2, x1 = 0, x2 = 1, and x3 = a. What are the possible value(s) of a?

Suppose that f (x) = 2x and g (x) = x +b. For what value(s) of b are f and g orthogonal with respect to the inner product in Example 5?

Suppose that f (x) = x2 and g (x) = x + b. For what value(s) of b are f and g orthogonal with respect to the inner product in Example 5?

For Exercises 1522, compute the norm with respect to the indicated inner product.      1 3 2      for the inner product given in Example 2.

For Exercises 1522, compute the norm with respect to the indicated inner product.       2 0 5      for the inner product given in Example 2 with weights t1 = 1, t2 = 5, and t3 = 2.

For Exercises 1522, compute the norm with respect to the indicated inner product. 3x 5 for the inner product given in Example 3 with x0 = 2, x1 = 1, and x2 = 4.

For Exercises 1522, compute the norm with respect to the indicated inner product.  x2 + x 4 for the inner product given in Example 3 with x0 = 0, x1 = 3, x2 = 2, and x3 = 6

For Exercises 1522, compute the norm with respect to the indicated inner product. x3 for the inner product given in Example 5.

For Exercises 1522, compute the norm with respect to the indicated inner product. xex for the inner product given in Example 5.

For Exercises 1522, compute the norm with respect to the indicated inner product. . A for A =  3 1 2 0 and A, B = tr(AT B).

For Exercises 1522, compute the norm with respect to the indicated inner product. A for A = 230 1 3 1 252 and A, B = tr(AT B).

For Exercises 2330, compute the indicated projection with respect to the given inner product. projuv for u = 1 2 1  , v = 3 4 2  , and the inner product given in Example 2.

For Exercises 2330, compute the indicated projection with respect to the given inner product. projuv for u = 2 5 2  , v = 1 3 0  , and the inner product given in Example 2 with weights t1 = 3, t2 = 1, and t3 = 4.

For Exercises 2330, compute the indicated projection with respect to the given inner product. projpq for p(x) = 3x + 2, q(x) = x + 1, and the inner product given in Example 3 with x0 = 1, x1 = 0, and x2 = 2.

For Exercises 2330, compute the indicated projection with respect to the given inner product. projpq for p(x) = x2 + 1, q(x) = 2x 3, and the inner product given in Example 3 with x0 = 1, x1 = 1, x2 = 2, and x3 = 5.

For Exercises 2330, compute the indicated projection with respect to the given inner product. proj f g for f (x) = x, g (x) = x2, and the inner product given in Example 5.

For Exercises 2330, compute the indicated projection with respect to the given inner product. proj f g for f (x) = sin(x), g (x) = 1 x2, and the inner product given in Example 5.

For Exercises 2330, compute the indicated projection with respect to the given inner product. projA B for A =  2 1 1 0 , B =  2 3 0 2  , and A, B = tr(AT B). 3

For Exercises 2330, compute the indicated projection with respect to the given inner product. . projA B for A =  3 4 1 3  , B =  1 5 2 1 , and A, B = tr(AT B). F

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. An orthogonal basis for R2 with respect to the inner product u, v = 3u1v1 + 2u2v2.

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. An orthogonal basis for P1 with respect to the inner product p, q =  1 0 p(x)q(x) dx.

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. A vector u such that u = 2 and u is orthogonal to  2 3  with respect to a weighted dot product of the form u, v = t1u1v1 + t2u2v2.

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. An inner product on P2 such that p = 3 for p(x) = x2 4x + 3.

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. A nonidentity matrix A such that u, v = uT Av is an inner product on R3.

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. A nonidentity matrix A such that u, v = uT Av is not an inner product on R3.

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. An inner product of your creation on P2.

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. An inner product of your own creation on C[0, 1].

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. A function p, q that is almost an inner product on P2: It satisfies (a)(c) of Definition 10.1, but not (d).

FIND AN EXAMPLE For Exercises 3140, find an example that meets the given specifications. A function A, B that is a poor attempt at an inner product on R33: It satisfies (a) of Definition 10.1, but not (b)(d).

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. If u, v = 3, then 2u, 4v=24.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. u + v2 = u2 + v2 for all u and v in V.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. If u and v are orthogonal with u = 3 and v = 4, then u + v = 5.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. If u = cv = 0 for a scalar c, then u = projuv.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. If {u, v} is an orthogonal set and c1 and c2 are scalars, then {c1u, c2v} is also an orthogonal set.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. . uv < (u,v) for all u and v in V.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. u vuv for all u and v in V.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. f, g =  1 1 x f (x)g (x) dx is an inner product on C[1, 1].

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. p, q = p(x0)q(x0) + p(x1)q(x1) is an inner product on P2 when x0 = x1.

TRUE OR FALSE For Exercises 4150, determine if the statement is true or false, and justify your answer. Here u and v are vectors in an inner product space V. If T : V R is a linear transformation, then u, v = T(u) T(v) is an inner product.

Complete Example 1. Prove that the weighted dot product on Rn given by u, v = t1u1v1 + t2u2v2 ++ tnunvn where t1, t2, ... , tn are all positive, is an inner product

Complete Example 3. Show that properties (a)(c) of Definition 10.1 are true for the inner product p, q = p(x0)q(x0) ++ p(xn)q(xn)

A weighted version of the inner product given in Example 3 is defined as follows: For p(x) and q(x) in Pn and distinct real numbers x0, x1, ... , xn, let p, q = t(x0)p(x0)q(x0) ++ t(xn)p(xn)q(xn) where t(x) takes positive values on x0, ... , xn. Show that p, q is an inner product on Pn.

Let f and g be continuous functions in C[1, 1]. Show that the weighted version of (3) given by f, g = 1 1 t(x) f (x)g (x) dx where t(x) > 0 is continuous for all x in [1, 1], defines an inner product on C[1, 1].

Let f and g be continuous functions in C[, ]. Show that f, g = 1 f (x)g (x) dx defines an inner product on C[, ].

Complete the proof of Theorem 10.6, by showing that parts (c) and (d) are true.

Prove that A, B = tr(AT B) is an inner product on R33.

For nonzero vectors u and v, show that there is equality in the CauchySchwarz inequality exactly when u = cv for some scalar c. (HINT: The key lies with the inequality (6).)

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. Prove that cv=|c|v for every v and scalar c.

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. Prove that v 0 for every v in V, with equality holding only for v = 0.

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. Prove that c1u1 ++ ckuk , w = c1u1, w++ ck uk , w

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. Prove that the zero vector 0 is orthogonal to all vectors in V.

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. Prove that if v = 0, then w = 1 v v satisfies w = 1.

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. Prove that projvu and u projvu are orthogonal.

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. For a fixed v in V, define Tv : V R by Tv(u) = u, v. Show that Tv is a linear transformation.

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. For a fixed v in V, define Tv : V V by Tv(u) = projvu. Show that Tv is a linear transformation.

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. Prove that if u and v are orthogonal, then the distance between u and v is u2 + v2.

For Exercises 5968, u, v, and w (and their subscripted associates) are vectors in an inner product space V. . Prove that u + v2 + u v2 = 2  u2 + v2

If S is a subspace of a finite-dimensional inner product space V, a vector v is orthogonal to S ifv, s = 0 for every vectorsin S. The set of all such vectors v is called the orthogonal complement of S and is denoted by S. In Exercises 6972, prove that the statement involving S is true. If S is a subspace, then so is S.

If S is a subspace of a finite-dimensional inner product space V, a vector v is orthogonal to S ifv, s = 0 for every vectorsin S. The set of all such vectors v is called the orthogonal complement of S and is denoted by S. In Exercises 6972, prove that the statement involving S is true. If S is a subspace, then  S = S.

If S is a subspace of a finite-dimensional inner product space V, a vector v is orthogonal to S ifv, s = 0 for every vectorsin S. The set of all such vectors v is called the orthogonal complement of S and is denoted by S. In Exercises 6972, prove that the statement involving S is true. If s is in S and s is in S, then s s 2 = s2 + s 2.

If S is a subspace of a finite-dimensional inner product space V, a vector v is orthogonal to S ifv, s = 0 for every vectorsin S. The set of all such vectors v is called the orthogonal complement of S and is denoted by S. In Exercises 6972, prove that the statement involving S is true. If S is a subspace, then S S = {0}.