Let V = Mat2(R) be the vector space (over R) of all 2 x 2 matrices with real entries (you do not need to prove this is a vector space) and let B, - {v. - (6 ") v* - (C ) • B; = {w, = (6 0) (C ) } - (: :) w- - (; ¿) } ,V3 ,V4 0 0 1 %3D W2 W3 = be bases for V (you do not need to show that they are bases). (i ') Fix a matrix A = EV and let T :V V be given by
Let V = Mat2(R) be the vector space (over R) of all 2 x 2 matrices with real entries (you do not need to prove this is a vector space) and let B, - {v. - (6 ") v* - (C ) • B; = {w, = (6 0) (C ) } - (: :) w- - (; ¿) } ,V3 ,V4 0 0 1 %3D W2 W3 = be bases for V (you do not need to show that they are bases). (i ') Fix a matrix A = EV and let T :V V be given by
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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please send handwritten solution for Q3
![3. Let V = Mat2(R) be the vector space (over R) of all 2 x 2 matrices with real
entries (you do not need to prove this is a vector space) and let
B, - {v. - (, 3)
(6 8)
- (C ?) }
, V2 =
, V3 =
0 0
B; = {w; = (6 )
(* )
-(: )
{(: :)-
W2 =
W3 =
W4 =
0 0
be bases for V (you do not need to show that they are bases).
(: ')
Fix a matrix A =
E V and let T :V →V be given by
T(M) = MA+ AM
for any M eV.
(i) Show that T is a linear transformation.
(ii) Find the change of basis matrix M from Bị to B2 and the change of basis
matrix N from B2 to B1 and verify that they are inverse to one another.
(iii) Give the coordinates of A in each of the bases B1, B2.
(iv) Find [T]B,.,B2, the matrix of T with respect to the bases B1, B2. 1'
4. Let V = R³ and let T: V → V be the linear map given by
T(r, y, z) = (2x + z, y – x+ z, 3:)
(you do not need to show that this is linear) and let
B = {v1 = (1, –1,0), v2 = (1,0,-1), v3 = (1,0,0)}
be a basis for V (you do not need to show that this is a basis).
(i) Verify that the vectors (0, 1,0), (1/2,0, 1/2), (1, -1,0) are each eigenvectors
of T and state their corresponding eigenvalues.
(ii) Find [T]B,B, the matrix of T with respect to the basis B.
(iii) Verify that the column vectors
are each eigenvectors of the matrix [TB.B you found in part (ii) and state
their corresponding eigenvalues.
5. Let A =
Find matrices C, D, where D is a diagonal matrix, such that
3 -5
A = CDC-1, and check your answer by calculating CDC-1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7995244-39eb-45ef-9c72-d3729c0dbdab%2Fd90cb968-9db2-4c8c-862e-6bb41abee67d%2Fa1brd1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Let V = Mat2(R) be the vector space (over R) of all 2 x 2 matrices with real
entries (you do not need to prove this is a vector space) and let
B, - {v. - (, 3)
(6 8)
- (C ?) }
, V2 =
, V3 =
0 0
B; = {w; = (6 )
(* )
-(: )
{(: :)-
W2 =
W3 =
W4 =
0 0
be bases for V (you do not need to show that they are bases).
(: ')
Fix a matrix A =
E V and let T :V →V be given by
T(M) = MA+ AM
for any M eV.
(i) Show that T is a linear transformation.
(ii) Find the change of basis matrix M from Bị to B2 and the change of basis
matrix N from B2 to B1 and verify that they are inverse to one another.
(iii) Give the coordinates of A in each of the bases B1, B2.
(iv) Find [T]B,.,B2, the matrix of T with respect to the bases B1, B2. 1'
4. Let V = R³ and let T: V → V be the linear map given by
T(r, y, z) = (2x + z, y – x+ z, 3:)
(you do not need to show that this is linear) and let
B = {v1 = (1, –1,0), v2 = (1,0,-1), v3 = (1,0,0)}
be a basis for V (you do not need to show that this is a basis).
(i) Verify that the vectors (0, 1,0), (1/2,0, 1/2), (1, -1,0) are each eigenvectors
of T and state their corresponding eigenvalues.
(ii) Find [T]B,B, the matrix of T with respect to the basis B.
(iii) Verify that the column vectors
are each eigenvectors of the matrix [TB.B you found in part (ii) and state
their corresponding eigenvalues.
5. Let A =
Find matrices C, D, where D is a diagonal matrix, such that
3 -5
A = CDC-1, and check your answer by calculating CDC-1.
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