3. Let TR4 →→ R³ be a linear transformation defined by T(x) = Ax where the matrix [ (a) Find T 0 T 0 2 0 A -1 2 0 and T -1 -1 0 1 1 3 0 -1 0 40 24 (b) Show that the columns of A span R³. Hence conclude that the image of T is R³. What is the rank of T? (c) What is the kernel (nullspace) of T? (d) Explain how we know that I has a right inverse, and that it does not have a left inverse. (e) Find values a, ß and y so that R: R³ → R¹ is a right inverse of T, where R(y) = By, and
3. Let TR4 →→ R³ be a linear transformation defined by T(x) = Ax where the matrix [ (a) Find T 0 T 0 2 0 A -1 2 0 and T -1 -1 0 1 1 3 0 -1 0 40 24 (b) Show that the columns of A span R³. Hence conclude that the image of T is R³. What is the rank of T? (c) What is the kernel (nullspace) of T? (d) Explain how we know that I has a right inverse, and that it does not have a left inverse. (e) Find values a, ß and y so that R: R³ → R¹ is a right inverse of T, where R(y) = By, and
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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QUESTION 3
Kindly answer correctly. Please sho all the necessary steps
Transcribed Image Text:3. Let TR4 →→ R³ be a linear transformation defined by T(x) = Ax where the matrix
[
(a) Find T
0
T
0
2
0
A
-1
2
0
and T
-1 -1 0
1
1
B =
3
0
-1
0
(b) Show that the columns of A span R³. Hence conclude that the image of T is R³. What
is the rank of T?
40
24
(c) What is the kernel (nullspace) of T?
(d) Explain how we know that I has a right inverse, and that it does not have a left inverse.
(e) Find values a, ß and y so that R: R³ → R¹ is a right inverse of T, where R(y) = By,
and
α
βγ
-2 -1 0
00
1
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