4. Let V = R³ and let T: V →V be the linear map given by T(r, y, 2) = (2x + z, y – x+ 2, 32) (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). %3D (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 1 are each eigenvectors of the matrix [T]B.B Yyou found in part (ii) and state their corresponding eigenvalues.

Advanced Engineering Mathematics
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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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.
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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