Consider the vector space R2. Define the mapping , by v, w = v1w1 v2w2, (5.1.15) for all

Use the standard inner product in R5 to determine the angle between the vectors v = (0, 2, 1, 4, 1) and w = (3, 1, 1, 0, 3).

Use the standard inner product in R4 to determine the angle between the vectors v = (1, 3, 1, 4) and w = (1, 1, 2, 1).

If f (x) = sin x and g(x) = x on [0,], use the function inner product (5.1.5) to determine the angle between f and g.

If f (x) = sin x and g(x) = 2 cos x + 4 on [0,/2], use the function inner product (5.1.5) to determine the angle between f and g.

Let m and n be positive real numbers. If f (x) = xm and g(x) = xn on an arbitrary interval [a, b], use the function inner product (5.1.5) to determine the angle between f and g in terms of a, b, m, and n.

If v = (2 + i, 3 2i, 4 + i) and w = (1 + i, 1 3i, 3 i), use the standard inner product in C3 to determine, v, w, ||v||, and ||w||.

If v = (6 3i, 4, 2 + 5i, 3i) and w = (i, 2i, 3i, 4i), use the standard inner product in C4 to determine, v, w, ||v||, and ||w||.

Let A =  a11 a12 a21 a22  and B =  b11 b12 b21 b22  be vectors in M2(R). Show that the mapping A, B = a11b11 does not define a valid inner product on M2(R). Which of the four properties of an inner product, if any, do hold? Justify your answer.

Referring to A and B in Problem 8, show that the mapping A, B = det(AB) does not define a valid inner product on M2(R). Which of the four properties of an inner product, if any, do hold? Justify your answer.

Referring to A and B in Problem 8, show that the mapping A, B = a11b22 + a12b21 + a21b12 + a22b11 does not define a valid inner product on M2(R). Which of the four properties of an inner product, if any, do hold? Justify your answer.

Referring to A and B in Problem 8, show that the mapping A, B = a11b11 + a12b12 + a21b21 + a22b22 (5.1.13) defines a valid inner product in M2(R).

For Problems 1213, use the inner product (5.1.13) in Problem 11 to determine A, B, ||A||, and ||B||. Also, determine the angle between the given matrices.

For Problems 1213, use the inner product (5.1.13) in Problem 11 to determine A, B, ||A||, and ||B||. Also, determine the angle between the given matrices.

Let p1(x) = a + bx and p2(x) = c + dx be vectors in P1(R). Determine a mapping p1, p2 that defines an inner product on P1(R).

Let V = C0[0, 1] and for f and g in V, consider the mapping  f, g =  1/2 0 f (x)g(x)dx. Does this define a valid inner product on V? Show why or why not.

Let V = C0[0, 1] and for f and g in V, consider the mapping  f, g =  1 0 x f (x)g(x)dx. Does this define a valid inner product on V? Show why or why not.

Let V = C0[1, 0] and for f and g in V, consider the mapping  f, g =  0 1 x f (x)g(x)dx. Does this define a valid inner product on V? Show why or why not.

Consider the vector space R2. Define the mapping , by v, w = 2v1w1 + v1w2 + v2w1 + 2v2w2 (5.1.14) for all vectors v = (v1,v2) and w = (w1,w2) in R2. Verify that Equation (5.1.14) defines an inner product on R2.

For Problems 1921, determine the inner product of the given vectors using (a) the inner product (5.1.14) in Problem 18, (b) the standard inner product in R2.v = (1, 0), w = (1, 2).

For Problems 1921, determine the inner product of the given vectors using (a) the inner product (5.1.14) in Problem 18, (b) the standard inner product in R2.v = (2, 1), w = (3, 6).

For Problems 1921, determine the inner product of the given vectors using (a) the inner product (5.1.14) in Problem 18, (b) the standard inner product in R2.v = (1, 2), w = (2, 1).

Consider the vector space R2. Define the mapping , by v, w = v1w1 v2w2, (5.1.15) for all vectors v = (v1,v2) and w = (w1,w2). Verify that all of the properties in Definition 5.1.3 except (1) are satisfied by (5.1.15). The mapping (5.1.15) is called a pseudo-inner productin R2 and, when generalized to R4, is of fundamental importance in Einsteins special relativity theory.

Using Equation (5.1.15) in Problem 22, determine all nonzero vectors satisfying v, v = 0. Such vectors are called null vectors.

Using Equation (5.1.15) in Problem 22, determine all vectors satisfying v, v < 0. Such vectors are called timelike vectors.

Using Equation (5.1.15) in Problem 22, determine all vectors satisfying v, v > 0. Such vectors are called spacelike vectors.

Make a sketch of R2 and indicate the position of the null, timelike, and spacelike vectors.

Consider the vector space R2. Define the mapping , by v, w = v1w1 v1w2 v2w1 + 4v2w2 for all vectors v = (v1,v2) and w = (w1,w2). Verify that , defines a valid inner product on R2.

Consider the vector space Rn, and let v = (v1,v2,...,vn) and w = (w1,w2,...,wn) be vectors in Rn. Complete the proof begun in Example 5.1.6 that the mapping , defined by v, w = k1v1w1 + k2v2w2 ++ knvnwn is a valid inner product on Rn if and only if the constants k1, k2,..., kn are all positive

Prove from the inner product axioms that, in any inner product space V, v, 0 = 0 for all v in V.

Prove from the inner product axioms that for all vectors u, v, and w in an inner product space V, we have u, v + w=u, v+u, w.

Use the previous exercise together with the inner product space axioms to derive a formula foru+v, w+x for vectors u, v, w, and x in an inner product space V.

Let V be an inner product space with vectors v and w with ||v|| = 3, ||w|| = 4, and v, w=2. Compute the following: (a) ||v + w||. (b) 3v + w, 2v + 3w. (c) w, 5v + 2w.

Let V be an inner product space with vectors v and w with ||v|| = 2, ||w|| = 6, and v, w = 3. Compute the following: (a) ||w 2v||. (b) ||2v + 5w||. (c) 2v w, 4w.

Let V be an inner product space with vectors u, v, and w with ||u|| = 1, ||v|| = 2, ||w|| = 3, u, v = 4, u, w = 5, and v, w = 0. Compute the following: (a) ||u + v + w||. (b) ||2u 3v w||. (c) u + 2w, 3u 3v + w.

Let V be a real inner product space. (a) Prove that for all v, w V, ||v + w||2 = ||v||2 + 2v, w + ||w||2. [Hint: ||v + w||2 = v + w, v + w.] (b) Two vectors v and w in an inner product space V are called orthogonal if v, w = 0. Use (a) to prove the general Pythagorean Theorem: If v and w are orthogonal in an inner product space V, then ||v + w||2 = ||v||2 + ||w||2.

Let V be a real inner product space. Prove that for all v, w in V, (a) ||v + w||2 ||v w||2 = 4v, w. (b) ||v + w||2 + ||v w||2 = 2(||v||2 + ||w||2).

Let V be a complex inner product space. Prove that for all v, w in V, ||v + w||2 = ||v||2 + 2Re(v, w) + ||v||2, where Re denotes the real part of a complex number.