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a *15. Given a = 6ER°, define T: R³ → R³ by T (x) = a x x. Prove that T is a linear trans- formation and give its standard matrix. Explain in the context of Proposition 4.5 why (T] is skew- symmetric.

a *15. Given a = 6ER°, define T: R³ → R³ by T (x) = a x x. Prove that T is a linear trans- formation and give its standard matrix. Explain in the context of Proposition 4.5 why (T] is skew- symmetric.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help me to solve #15. Thank you so much!!

Proposition 4.5 Let A be an m x n matrix, x e R", and y e R". Then Ax · y = x. A'y. (On the left, we take the dot product of vectors in R"; on the right, of vectors in R".) Remark You might remember this: To move the matrix "across the dot product," you must transpose it. Proof We just calculate, using the formula for the transpose of a product and, as usual, associativity: Ax - y = (Ax)"y = (xA")y = x"(A"y) = x · A"y. I
Transcribed Image Text:Proposition 4.5 Let A be an m x n matrix, x e R", and y e R". Then Ax · y = x. A'y. (On the left, we take the dot product of vectors in R"; on the right, of vectors in R".) Remark You might remember this: To move the matrix "across the dot product," you must transpose it. Proof We just calculate, using the formula for the transpose of a product and, as usual, associativity: Ax - y = (Ax)"y = (xA")y = x"(A"y) = x · A"y. I
a *15. Given a =beR', define T: R³ → R' by T (x) = a x x. Prove that T is a linear trans- formation and give its standard matrix. Explain in the context of Proposition 4.5 why (T] is skew- symmetric.
Transcribed Image Text:a *15. Given a =beR', define T: R³ → R' by T (x) = a x x. Prove that T is a linear trans- formation and give its standard matrix. Explain in the context of Proposition 4.5 why (T] is skew- symmetric.
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