(the trace of A). Prove that T is a linear transformation. Let A and B be to matrices in M, and let the entries of A, B be denoted a, and b;, respectively, for 1 i,jsn. By matrix addition, 'n, we know that the entries of the matrix A + B are of the form aj + bị and associative property of addition, we have which of the fol ---Select--- ai · bj v for 1 si, j s n. By the definition of T and the commutative T(A + B) = (a, ·b,) + -… + %3D aj + bị ("q ."e) - = (a, + ... + a,) + (b, + .… + = T(A) + T(B) hq . he aj + bj T(A + B) = (a, + b;)· + ... + (a, + b,) ai + b = (a, + ... + a,) + (b1 = T(A) + T(B) T(A + B) = (a11 + b11) - + (ann bo) = (a,1 + ** + ann) + (b,1 + + bon) = T(A) + T(B) T(A + B) = (a11 b11) + … + (ann :bn) = (a,, + … + ann) + (b,1 + . + bon) = T(A) + T(B) Now, let c be a scalar. By scalar multiplication, we know that the entries of the matrix CA are of the form + aj v, 1si,jsn. By the definition of T and the distributive property of multiplication, we have which of the following? T(CA) = ca, + ** + ca, = c(a, + ... + a,) = cT(A) T(CA) = ca11 + ... + ca n = c(a,, + . + a = cT(A) T(CA) = ca, ca, = c(a, + ... + a) = cT(A) T(CA) = ca1 · can = c(a11 = CT(A) + ... + a nn)

(the trace of A). Prove that T is a linear transformation. Let A and B be to matrices in M, and let the entries of A, B be denoted a, and b;, respectively, for 1 i,jsn. By matrix addition, 'n, we know that the entries of the matrix A + B are of the form aj + bị and associative property of addition, we have which of the fol ---Select--- ai · bj v for 1 si, j s n. By the definition of T and the commutative T(A + B) = (a, ·b,) + -… + %3D aj + bị ("q ."e) - = (a, + ... + a,) + (b, + .… + = T(A) + T(B) hq . he aj + bj T(A + B) = (a, + b;)· + ... + (a, + b,) ai + b = (a, + ... + a,) + (b1 = T(A) + T(B) T(A + B) = (a11 + b11) - + (ann bo) = (a,1 + + ann) + (b,1 + + bon) = T(A) + T(B) T(A + B) = (a11 b11) + … + (ann :bn) = (a,, + … + ann) + (b,1 + . + bon) = T(A) + T(B) Now, let c be a scalar. By scalar multiplication, we know that the entries of the matrix CA are of the form + aj v, 1si,jsn. By the definition of T and the distributive property of multiplication, we have which of the following? T(CA) = ca, + + ca, = c(a, + ... + a,) = cT(A) T(CA) = ca11 + ... + ca n = c(a,, + . + a = cT(A) T(CA) = ca, ca, = c(a, + ... + a) = cT(A) T(CA) = ca1 · can = c(a11 = CT(A) + ... + a nn)

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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