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Math / Linear Algebra 4 / Chapter 6.1 / Problem 8

Linear Algebra | 4th Edition | ISBN: 9780130084514 | Authors: Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence

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Provide reasons why each of the following is not an inner product on the given vector

Chapter 6, Problem 8

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QUESTION:

Provide reasons why each of the following is not an inner product on the given vector spaces. (a) ((a.b).(c.d)) =ac bd on R2 . (b) (A, B) = tv(A 4- B) on U2x2(R). (c) (f(x),g(x)) = /J f'(t)g(t)dt on P(R), where ' denotes differentiation.

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QUESTION:

Provide reasons why each of the following is not an inner product on the given vector spaces. (a) ((a.b).(c.d)) =ac bd on R2 . (b) (A, B) = tv(A 4- B) on U2x2(R). (c) (f(x),g(x)) = /J f'(t)g(t)dt on P(R), where ' denotes differentiation.

ANSWER:

Step 1 of 4

Definition

Let be a vector space over . An inner product on is a function that assigns, to every ordered pair of vectors and in , a scalar in , denoted , such that for all , and in and all in , the following hold

(a) .

(b)

(c) where the bar denotes complex conjugation.

(d) If , then is a positive real number.

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