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Let 0 1 1 -1 0 A 0 1 1 1 0 1 2372 0 0 2 0 (a) Find an ordered basis for ran(A), what is dim(ran(A))? (b) A defines a linear transformation T : Rn → Rk by T(x) = Añ. What are n and k? (c) Use the Rank-Nullity theorem to find dim(ker(A)). Then, find a basis for ker(A). (d) Add vectors to the ordered basis for ran(A) you found in (a) to get an ordered basis B for Rk where k is as in (b) (e) Let be the standard basis for Rk. Find Pe→B. (f) For each of the following, find the set of solutions if there are any, if there is no solutions, explain why. Ax= 333 3/2 2 Ay = Az = -1/2 3/2 1002

Let 0 1 1 -1 0 A 0 1 1 1 0 1 2372 0 0 2 0 (a) Find an ordered basis for ran(A), what is dim(ran(A))? (b) A defines a linear transformation T : Rn → Rk by T(x) = Añ. What are n and k? (c) Use the Rank-Nullity theorem to find dim(ker(A)). Then, find a basis for ker(A). (d) Add vectors to the ordered basis for ran(A) you found in (a) to get an ordered basis B for Rk where k is as in (b) (e) Let be the standard basis for Rk. Find Pe→B. (f) For each of the following, find the set of solutions if there are any, if there is no solutions, explain why. Ax= 333 3/2 2 Ay = Az = -1/2 3/2 1002

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 26EQ
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Let 1 0 1 2 Ο 1 −1 0 3 0 A 0 1 1 -1 2 - 1 012 0 (a) Find an ordered basis for ran(A), what is dim(ran(A))? (b) A defines a linear transformation T : Rn → Rk by T(Ã) = Ax. What are n and k? (c) Use the Rank-Nullity theorem to find dim(ker(A)). Then, find a basis for ker(A). (d) Add vectors to the ordered basis for ran(A) you found in (a) to get an ordered basis B for Rk where k is as in (b) (e) Let be the standard basis for Rk. Find Pe→ß. (f) For each of the following, find the set of solutions if there are any, if there is no solutions, explain why. [4 3 3/2 2 0 Ax = Ay = Az= 3 -1/2 0 3 3/2 2
Transcribed Image Text:Let 1 0 1 2 Ο 1 −1 0 3 0 A 0 1 1 -1 2 - 1 012 0 (a) Find an ordered basis for ran(A), what is dim(ran(A))? (b) A defines a linear transformation T : Rn → Rk by T(Ã) = Ax. What are n and k? (c) Use the Rank-Nullity theorem to find dim(ker(A)). Then, find a basis for ker(A). (d) Add vectors to the ordered basis for ran(A) you found in (a) to get an ordered basis B for Rk where k is as in (b) (e) Let be the standard basis for Rk. Find Pe→ß. (f) For each of the following, find the set of solutions if there are any, if there is no solutions, explain why. [4 3 3/2 2 0 Ax = Ay = Az= 3 -1/2 0 3 3/2 2
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