MULTIPLE CHOICE( as answering as a, b,c,d,e would be enough, not need for proovi

Question

MULTIPLE CHOICE( as answering as a, b,c,d,e would be enough, not need for prooving)

1.In the real inner product space P 3 of polynomials with the integral inner product

a.{1, x}

b.{1, x2}

c.{x, x2}

d.{1, 2x + 3x2}

e.{x, 2 - 3x}

2.Let U be any n-dimensional real inner-product space with arbitrary basis B = {u 1, u 2, ..., u n}

a.For u U, then u = <u, u 1>u 1 + <u, u 2>u 2 + ... + <u, u n>u n.

b.If u = a1u 1 + a2u 2 + ... + anu n, and v = b1u 1 + b2u 2 + ... + bnu n, then <u,v> = a1b1 + a2b2 + ... + anbn

c.With u and v given in choice "b" above, <u, v> = 0, if and only if, u or v = 0.

d.If B is an orthonormal basis, choices "a", "b", and "c" are all true.

e.If B is an orthonormal basis, choices "a" and "b" are both true, but "c" is false.

a.True

b.False

4.In an arbitrary complex inner-product space V which of the following is not always true?

a.<u + v, w> = <u, w> + <v, w>

b.| <u, v> |2 <u, u> <v,v>

c.<u, v> = <u, v>

d.<0, u> = 0

5.Which one of the following statements is not true?

b.In any inner-product space V, ||v|| = | | ||v||, for any vector v.

c.In any inner-product space V, ||u|| = <u, u>½, for any vector u.

d.In any inner-product space V, <u, u> 0, for any vector u.

MULTIPLE CHOICE( as answering as a, b,c,d,e would be enough, not need for prooving) 1.In the real inner product space P 3 of polynomials with the integral inner product = p(x)q(x) dx which of the following pairs of vectors is orthogonal? a.{1, x} b.{1, x2} c.{x, x2} d.{1, 2x + 3x2} e.{x, 2 - 3x} 2.Let U be any n-dimensional real inner-product space with arbitrary basis B = {u 1, u 2, ..., u n} a.For u ? U, then u = u 1 + u 2 + ... + u n. b.If u = a1 u 1 + a2 u 2 + ... + an u n, and v = b1 u 1 + b2 u 2 + ... + bn u n, then = a1b1 + a2b2 + ... + anbn c.With u and v given in choice b above, = 0, if and only if, u or v = 0. d.If B is an orthonormal basis, choices a, b, and c are all true. e.If B is an orthonormal basis, choices a and b are both true, but c is false. 3.The set of vectors B = {(-1, -1, 1, 1), (1, 0, 0, 0), (0, 1, 0, 0), (-1, -1, 1, -1,)} is an orthonormal basis for Euclidean 4-space 4. a.True b.False 4.In an arbitrary complex inner-product space V which of the following is not always true? a.

Explanation / Answer

1. e 2. e 3. b 4. b 5. a

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