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Let 1 0 1 2 0 -1 -1 1 0 0 1 1 0 1 1 and let T: R6 →→ R5 be defined by T(x) = At. (a) Find a basis for ran(T). What is dim(ran(T))? (b) Use the Rank-Nullity theorem to find dim(ker(T)). (c) Find a basis for ker(T), i.e. a linearly independent set of vectors from ker(T) of size dim(ker(T)). (d) What are the solution sets to A = Az = -1 -2 2 2 2 0 -3 21022 3600 0 0 and Az = [0] 0 1? 1

Let 1 0 1 2 0 -1 -1 1 0 0 1 1 0 1 1 and let T: R6 →→ R5 be defined by T(x) = At. (a) Find a basis for ran(T). What is dim(ran(T))? (b) Use the Rank-Nullity theorem to find dim(ker(T)). (c) Find a basis for ker(T), i.e. a linearly independent set of vectors from ker(T) of size dim(ker(T)). (d) What are the solution sets to A = Az = -1 -2 2 2 2 0 -3 21022 3600 0 0 and Az = [0] 0 1? 1

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 2 0 -1 1 0 1 0 1 and let T : R6 → R5 be defined by T(x) = Ax. (a) Find a basis for ran(T). What is dim(ran(T))? (b) Use the Rank-Nullity theorem to find dim(ker(T)). (c) Find A = Ax = basis for ker(T), i.e. a linearly independent set of vectors from ker(T) of size dim(ker (T)). (d) What are the solution sets to 1 2 1 -2 2 -1 2 1 0 2 0 -3 1 0 2 0 1 0 2 0 0 36 and Ax=1 ?
Transcribed Image Text:Let 1 0 2 0 -1 1 0 1 0 1 and let T : R6 → R5 be defined by T(x) = Ax. (a) Find a basis for ran(T). What is dim(ran(T))? (b) Use the Rank-Nullity theorem to find dim(ker(T)). (c) Find A = Ax = basis for ker(T), i.e. a linearly independent set of vectors from ker(T) of size dim(ker (T)). (d) What are the solution sets to 1 2 1 -2 2 -1 2 1 0 2 0 -3 1 0 2 0 1 0 2 0 0 36 and Ax=1 ?
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