Question 2 1. Find the basis of the range and the kernel of the following linear trans- formations. (a) T: R3 R² such that T() = -1-2-3 -2-4-6 Z. [5] -x+y (b) T: R3 R³, such that T = 0 [4] -x+y 2 1-1 3 (c) T: RR3 such that T() = I. [5] 2 1 0 03 -6 2. Let PVR and Q: V→R be linear transformations, where V is a vector space. Define T: VR2 by T(v) = (P(v), Q(v)). (a) Show that T is a linear transformation. [4] (b) Show that ker T = ker Poker Q, the set of vectors in both ker P and ker Q. [5] 3. Let T (R2,S) →> (R2, 7) be defined by T x1-x2 Find the matrix M representing T when. (a) S= (b) S= = {(~)·()}• T = {(~)·()}· T= = {(²)·({}) } = {( -3)·(~})} (c) In each case (a) and (b) above, calculate T .T= by using M. [6] [6] ((²)) directly and [8]
Question 2 1. Find the basis of the range and the kernel of the following linear trans- formations. (a) T: R3 R² such that T() = -1-2-3 -2-4-6 Z. [5] -x+y (b) T: R3 R³, such that T = 0 [4] -x+y 2 1-1 3 (c) T: RR3 such that T() = I. [5] 2 1 0 03 -6 2. Let PVR and Q: V→R be linear transformations, where V is a vector space. Define T: VR2 by T(v) = (P(v), Q(v)). (a) Show that T is a linear transformation. [4] (b) Show that ker T = ker Poker Q, the set of vectors in both ker P and ker Q. [5] 3. Let T (R2,S) →> (R2, 7) be defined by T x1-x2 Find the matrix M representing T when. (a) S= (b) S= = {(~)·()}• T = {(~)·()}· T= = {(²)·({}) } = {( -3)·(~})} (c) In each case (a) and (b) above, calculate T .T= by using M. [6] [6] ((²)) directly and [8]
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 2
1. Find the basis of the range and the kernel of the following linear trans-
formations.
(a) T: R3 R² such that T() =
-1-2-3
-2-4-6
Z.
[5]
-x+y
(b) T: R3 R³, such that T
=
0
[4]
-x+y
2
1-1 3
(c) T: RR3 such that T() =
I.
[5]
2
1 0
03
-6
2. Let PVR and Q: V→R be linear transformations, where V is
a vector space. Define T: VR2 by T(v) = (P(v), Q(v)).
(a) Show that T is a linear transformation.
[4]
(b) Show that ker T = ker Poker Q, the set of vectors in both ker P
and ker Q.
[5]
3. Let T (R2,S) →> (R2, 7) be defined by T
x1-x2
Find the matrix M representing T when.
(a) S=
(b) S=
= {(~)·()}• T = {(~)·()}·
T=
= {(²)·({}) } = {( -3)·(~})}
(c) In each case (a) and (b) above, calculate T
.T=
by using M.
[6]
[6]
((²))
directly and
[8]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61df6a42-13ad-4413-8d97-1d07dfc3ca7e%2Fa82b8b47-0e11-4a57-87fc-e1fd4491c3a8%2Fxqeyl1g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 2
1. Find the basis of the range and the kernel of the following linear trans-
formations.
(a) T: R3 R² such that T() =
-1-2-3
-2-4-6
Z.
[5]
-x+y
(b) T: R3 R³, such that T
=
0
[4]
-x+y
2
1-1 3
(c) T: RR3 such that T() =
I.
[5]
2
1 0
03
-6
2. Let PVR and Q: V→R be linear transformations, where V is
a vector space. Define T: VR2 by T(v) = (P(v), Q(v)).
(a) Show that T is a linear transformation.
[4]
(b) Show that ker T = ker Poker Q, the set of vectors in both ker P
and ker Q.
[5]
3. Let T (R2,S) →> (R2, 7) be defined by T
x1-x2
Find the matrix M representing T when.
(a) S=
(b) S=
= {(~)·()}• T = {(~)·()}·
T=
= {(²)·({}) } = {( -3)·(~})}
(c) In each case (a) and (b) above, calculate T
.T=
by using M.
[6]
[6]
((²))
directly and
[8]
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