Solution Found!
em 10. Let T : n m be a linear transformation such that T
Chapter 1, Problem 33E(choose chapter or problem)
PROBLEM 33EVerify the uniqueness of A in Theorem 10. Let T : ?n? ?m be a linear transformation such that T (x) = Bx for some m × n matrix B. Show that if A is the standard matrix for T , then A = B. [Hint: Show that A and B have the same columns.]Theorem 10:Let T : ?n? ?m be a linear transformation. Then there exists a unique matrix A such that for all x in ?nIn fact, A is the m × n matrix whose j th column is the vector where ej is the j th column of the identity matrix in ?n:
Questions & Answers
QUESTION:
PROBLEM 33EVerify the uniqueness of A in Theorem 10. Let T : ?n? ?m be a linear transformation such that T (x) = Bx for some m × n matrix B. Show that if A is the standard matrix for T , then A = B. [Hint: Show that A and B have the same columns.]Theorem 10:Let T : ?n? ?m be a linear transformation. Then there exists a unique matrix A such that for all x in ?nIn fact, A is the m × n matrix whose j th column is the vector where ej is the j th column of the identity matrix in ?n:
ANSWER:SolutionStep 1Let T : n m be a linear transformation such that T (x) = Bx for some m × n matrix B.Let x = In x where In = (e1 , e2 , ……… , en) and use the linearity of T is given by T(x) = T(In x) = T (e1 , e2 , ……… , en)x = T (x1e1 + x1e1 +................... +xnen ) = x1T(e1) + x2T(e2) + ………………. + xnT(en) Suppose A be a unique Matrix whose j th column is the vector where ej is the j th column of the identity matrix in n: Now, Hence, It is verified the uniqueness of A.