EXERCISE Let C = (1, 2, 3, 4) be the canonical basis of R¹, where e₁=(1,0,0,0), z = (0,1,0,0), es= (0,0,1,0), e = (0,0,0,1). Consider the endomorphism f: R -R such that f(z,y,z,t) = (y, z, z,t). (a) Write the matrix M(f) associated to f with respect to the canonical basis both in the domain and codomain. (b) Find a basis of the eigenspace relative to the eigenvalue -1. (e) Prove or disprove: f is simple (that is, diagonalizable). (d) Write down all the elements of the set I= {ve R¹ : f(v) = (1,2,3,4)}. (e) Given the basis of R' B= (₁, 12, 13, 14), where =(0,0,0,2), (1,0,0,2), 3=(-1,-1,0,0), 4(0,0,1,-2), Write the matrix M(f) associated to f with respect to the basis B both in the domain and codomain. 01 a) M₂ (g) = 10 O O O b) Basis for Eg(1) = ((1,-1,0,0)) c) is diagonalizable because it is symmetric - In particular, the eigenvalves are d₁=-1, ma (-1) = mg (1) = 1 and d₂=1, ma (1) = mg (1) = 3) d) I = {(2,1,3,4)} 3) MB ( 1200 0-1 OC 0-1 10 bol
EXERCISE Let C = (1, 2, 3, 4) be the canonical basis of R¹, where e₁=(1,0,0,0), z = (0,1,0,0), es= (0,0,1,0), e = (0,0,0,1). Consider the endomorphism f: R -R such that f(z,y,z,t) = (y, z, z,t). (a) Write the matrix M(f) associated to f with respect to the canonical basis both in the domain and codomain. (b) Find a basis of the eigenspace relative to the eigenvalue -1. (e) Prove or disprove: f is simple (that is, diagonalizable). (d) Write down all the elements of the set I= {ve R¹ : f(v) = (1,2,3,4)}. (e) Given the basis of R' B= (₁, 12, 13, 14), where =(0,0,0,2), (1,0,0,2), 3=(-1,-1,0,0), 4(0,0,1,-2), Write the matrix M(f) associated to f with respect to the basis B both in the domain and codomain. 01 a) M₂ (g) = 10 O O O b) Basis for Eg(1) = ((1,-1,0,0)) c) is diagonalizable because it is symmetric - In particular, the eigenvalves are d₁=-1, ma (-1) = mg (1) = 1 and d₂=1, ma (1) = mg (1) = 3) d) I = {(2,1,3,4)} 3) MB ( 1200 0-1 OC 0-1 10 bol
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
I don’t understand in this question part e
How did the teacher get the matrix my columns are coming different
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EXERCISE
Let C = (e₁, 02, 03, ea) be the canonical basis of R4, where
e₁ = (1,0,0,0), e₂ = (0,1,0,0), e3= (0,0,1,0), e4 = (0,0,0,1).
Consider the endomorphism f: RR such that f(x, y, z, t) = (y, z, z, t).
(a) Write the matrix M(f) associated to f with respect to the canonical basis both in the domain
and codomain.
(b) Find a basis of the eigenspace relative to the eigenvalue -1.
(c) Prove or disprove: f is simple (that is, diagonalizable).
(d) Write down all the elements of the set I= (vER: f(v) = (1,2,3,4)}.
(e) Given the basis of R4 B= (v1, 12, 13, 14), where
$
v₁ = (0,0,0,2), 12 = (1,0,0,2), 3= (-1,-1,0,0), v4 = (0,0,1,-2),
Write the matrix MB (f) associated to f with respect to the basis B both in the domain and
codomain.
a) Me (g) =
Q
F4
R
b) Basis for Eg(1) = ((1,-1,0,0))
c) is diagonalizable because it is
symmetric - In particular, the
eigenvalves are d₁=-1₁ m₂ (-1) = mg (-1)
= 1 and d₂=1, ma (1) = mg (1) = 3)
d) I = {(2,1,3,4]}
(
O -1
MB 9-
F
%
5
I
0100
- 0 0 0
T
1
G
NTTO
0-1
O
00-0
&
6
0.0
000-
00-0
оо
- 200
Y
1
7
H
U
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8
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Transcribed Image Text:W
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EXERCISE
Let C = (e₁, 02, 03, ea) be the canonical basis of R4, where
e₁ = (1,0,0,0), e₂ = (0,1,0,0), e3= (0,0,1,0), e4 = (0,0,0,1).
Consider the endomorphism f: RR such that f(x, y, z, t) = (y, z, z, t).
(a) Write the matrix M(f) associated to f with respect to the canonical basis both in the domain
and codomain.
(b) Find a basis of the eigenspace relative to the eigenvalue -1.
(c) Prove or disprove: f is simple (that is, diagonalizable).
(d) Write down all the elements of the set I= (vER: f(v) = (1,2,3,4)}.
(e) Given the basis of R4 B= (v1, 12, 13, 14), where
$
v₁ = (0,0,0,2), 12 = (1,0,0,2), 3= (-1,-1,0,0), v4 = (0,0,1,-2),
Write the matrix MB (f) associated to f with respect to the basis B both in the domain and
codomain.
a) Me (g) =
Q
F4
R
b) Basis for Eg(1) = ((1,-1,0,0))
c) is diagonalizable because it is
symmetric - In particular, the
eigenvalves are d₁=-1₁ m₂ (-1) = mg (-1)
= 1 and d₂=1, ma (1) = mg (1) = 3)
d) I = {(2,1,3,4]}
(
O -1
MB 9-
F
%
5
I
0100
- 0 0 0
T
1
G
NTTO
0-1
O
00-0
&
6
0.0
000-
00-0
оо
- 200
Y
1
7
H
U
(
8
J
9
K
O
O
F10
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