For each of the following linear operators T on a vector space V and ordered bases ẞ, compute [T], and determine whether ẞ is a basis consisting of eigenvectors of T. a (a) V = R², T = (199 10a- 6b and B = " {(2), (3)} 17a10b (b) VP1(R), T(a + b) = (6a-6b) + (12a - 11b)x, and B={3+4x, 2+3x} (c) V = R³, T a - 3a+26 2c\ -4a-3b+2c TO) -(+ C and " B = (d) VP2(R), T(a + bx + cx²) = (-4a+26-2c) - (7a+3b+7c)x+(7a+b+5c)x², and ẞ {x x2, -1+x2,-1-x+x2} = -
For each of the following linear operators T on a vector space V and ordered bases ẞ, compute [T], and determine whether ẞ is a basis consisting of eigenvectors of T. a (a) V = R², T = (199 10a- 6b and B = " {(2), (3)} 17a10b (b) VP1(R), T(a + b) = (6a-6b) + (12a - 11b)x, and B={3+4x, 2+3x} (c) V = R³, T a - 3a+26 2c\ -4a-3b+2c TO) -(+ C and " B = (d) VP2(R), T(a + bx + cx²) = (-4a+26-2c) - (7a+3b+7c)x+(7a+b+5c)x², and ẞ {x x2, -1+x2,-1-x+x2} = -
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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Section 5.1: 3(d)only!
![3. For each of the following linear operators T on a vector space V and
ordered bases ẞ, compute [T]s, and determine whether ẞ is a basis
consisting of eigenvectors of T.
a
(a) V = R², T
= (199
-
10a 6b
17a10b
, and 3 =
{(2), (3)}
(b) VP1(R), T(a + bx)
B = {3+ 4x, 2+3x}
a
(c) V R3, Tb
=
(O) -(
C
=
-
(6a 6b)+(12a - 11b)x, and
3a+2b 2c\
-
-4a-3b+2c, and
-{(0)·(9)·0))
2
(d) VP2(R), T(a + bx + cx²)
=
-C
(-4a2b2c) - (7a+3b+7c)x + (7a+b+5c)x²,
and ẞ= {x-x², -1 + x², -1- x + x ² }
(e) VP3(R), T(a + bx + cx² + dx³)
=
-d+(-c+d)x + (a + b − 2c)x² + (b + c = 2d) x³,
-
and ẞ {1x+x3,1 + x², 1, x + x2}
=
b)
-
-
-7a-4b+4c-4d b
-8a 4b+5c 4d d
-
-
(f) V = M2x2 (R), T ( a b )
c
¿).
and
β
- {( ) ( ) ( ) (9)}
1 0
"
"
2
0
2
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd3be9fc5-49dc-4987-93dc-30b0d6519ab7%2F91affbdf-2932-4a4a-a6cd-fec797c317c6%2Fo8p0wrk_processed.png&w=3840&q=75)
Transcribed Image Text:3. For each of the following linear operators T on a vector space V and
ordered bases ẞ, compute [T]s, and determine whether ẞ is a basis
consisting of eigenvectors of T.
a
(a) V = R², T
= (199
-
10a 6b
17a10b
, and 3 =
{(2), (3)}
(b) VP1(R), T(a + bx)
B = {3+ 4x, 2+3x}
a
(c) V R3, Tb
=
(O) -(
C
=
-
(6a 6b)+(12a - 11b)x, and
3a+2b 2c\
-
-4a-3b+2c, and
-{(0)·(9)·0))
2
(d) VP2(R), T(a + bx + cx²)
=
-C
(-4a2b2c) - (7a+3b+7c)x + (7a+b+5c)x²,
and ẞ= {x-x², -1 + x², -1- x + x ² }
(e) VP3(R), T(a + bx + cx² + dx³)
=
-d+(-c+d)x + (a + b − 2c)x² + (b + c = 2d) x³,
-
and ẞ {1x+x3,1 + x², 1, x + x2}
=
b)
-
-
-7a-4b+4c-4d b
-8a 4b+5c 4d d
-
-
(f) V = M2x2 (R), T ( a b )
c
¿).
and
β
- {( ) ( ) ( ) (9)}
1 0
"
"
2
0
2
1
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