1. Let A = 3 ε = 1 and consider the bases 1 0 {69-663-69} -{[]+9+*+]} {B 0 of the vector space M2x2 of 2 x 2 matrices. (a) Find [12] and [A]ɛ. (Recall, for example, [12]e is the coordinate vector of I2 relative to the ordered basis & for M2x2.) (b) Find [12] and [A]B. and B = 1 (c) Find a basis C of M2x2 such that [A]c =
1. Let A = 3 ε = 1 and consider the bases 1 0 {69-663-69} -{[]+9+*+]} {B 0 of the vector space M2x2 of 2 x 2 matrices. (a) Find [12] and [A]ɛ. (Recall, for example, [12]e is the coordinate vector of I2 relative to the ordered basis & for M2x2.) (b) Find [12] and [A]B. and B = 1 (c) Find a basis C of M2x2 such that [A]c =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![1. Let A =
3 1
34
-1
9
and consider the bases
To 1] [o o To
0
0
²-{ · 1)
P 1.6 }}
=
{b
{[169]]]])}
00 00 10 01
0 -1
;}}
0
0
of the vector space M2×2 of 2 x 2 matrices. 1
(a) Find [I₂]ɛ and [A]ɛ. (Recall, for example, [12]ɛ is the coordinate vector of 12 relative to
the ordered basis & for M2×2.)
(b) Find [I2] and [A]B.
(c) Find a basis C of M2x2 such that
[A]c
and B
=
B =
1
1
1
1
2
(d) Find a matrix C such that C[B]ß = [B]c for all B in M₂×2.
(e) Find a matrix D such that D[B]c = [B]ɛ for all B in M₂×2.
(f) Find a matrix F such that F[B]ß = [B]ɛ for all B in M2×2.
(g) Draw a diagram relating the linear transformations corresponding to the matrices F, C
and D.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F671c4bae-6499-423f-ab48-7cf20982a060%2F31b0f59b-c8fb-49cc-959f-24a8d4c01858%2Fgttyo2n_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let A =
3 1
34
-1
9
and consider the bases
To 1] [o o To
0
0
²-{ · 1)
P 1.6 }}
=
{b
{[169]]]])}
00 00 10 01
0 -1
;}}
0
0
of the vector space M2×2 of 2 x 2 matrices. 1
(a) Find [I₂]ɛ and [A]ɛ. (Recall, for example, [12]ɛ is the coordinate vector of 12 relative to
the ordered basis & for M2×2.)
(b) Find [I2] and [A]B.
(c) Find a basis C of M2x2 such that
[A]c
and B
=
B =
1
1
1
1
2
(d) Find a matrix C such that C[B]ß = [B]c for all B in M₂×2.
(e) Find a matrix D such that D[B]c = [B]ɛ for all B in M₂×2.
(f) Find a matrix F such that F[B]ß = [B]ɛ for all B in M2×2.
(g) Draw a diagram relating the linear transformations corresponding to the matrices F, C
and D.
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