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Let 0 1 2 0 -1 0 3 0 A = 1 1 −1 2 1 0 1 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).

Let 0 1 2 0 -1 0 3 0 A = 1 1 −1 2 1 0 1 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).

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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Let [1 0 1 2 01 -1 0 0 A = 0 1 -1 2 1 0 1 0 (a) Find an ordered basis for ran(A), what is dim(ran(A))? (b) A defines a linear transformation T : R^ → Rk by T(x) = 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→. (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 1902
Transcribed Image Text:Let [1 0 1 2 01 -1 0 0 A = 0 1 -1 2 1 0 1 0 (a) Find an ordered basis for ran(A), what is dim(ran(A))? (b) A defines a linear transformation T : R^ → Rk by T(x) = 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→. (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 1902
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