Answered: Let 0 1 2 0 -1 0 3 0 A = 1 1 −1 2 1 0 1… | bartleby

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 20EQ

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Question

Please help with solution + steps for parts a,b,c

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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