Let T : M2×2(R) → M2×2(R) be the linearoperator given byTa bc d =16a 5b + 11c11b + 5c 16d,where M2×2(R) denotes the vector space of 2 × 2-matrices over R.Let B =" 1 00 0 ,0 10 0 ,0 01 0 ,0 00 1 #denotethe standard basis of M2×2(R).(a) Find [T]B. Determine the characteristic polynomial of [T]B, i.e.P[T]B(λ). Find the eigenvalues of [T]B and determine the algebraic multiplicity of each eigenvalue.(Hint: Eigenvalues are −6 and 16.)(b) Determine a basis for each eigenspace E[T]B(λ) of [T]B and determine the geometric multiplicity of each eigenvalue.(c) Use parts a and b to determine the characteristic polynomialPT (λ) of T, eigenvalues of T and a basis for each eigenspaceET (λ) of T. Show that eigenvectors of T form a basis B0forM2×2(R).(d) Find the representation [T]B0 of T relative to the basis B0. Use[T]B0 to verify that T is diagonalizable and T is an isomorphism. Let T: M2x2(R) → M2x2(R) be the linearoperator given bya b16a56 + 11c]11b + 5c16dwhere M2x2(R) denotes the vector space of 2 x 2-matrices over R.0 00 10 0Let B =denotethe standard basis of M2x2(R).(a) Find [T]B. Determine the characteristic polynomial of [T]B, i.e.Pr, (A). Find the eigenvalues of [T]B and determine the alge-braic multiplicity of each eigenvalue.(Hint: Eigenvalues are -6 and 16.)(b) Determine a basis for each eigenspace ET, (A) of [T]B and de-termine the geometric multiplicity of each eigenvalue.(c) Use parts a and b to determine the characteristic polynomialPr (X) of T, eigenvalues of T and a basis for each eigenspaceET(A) of T. Show that eigenvectors of T form a basis B' forM2x2(R).(d) Find the representation [T]g of T relative to the basis B'. Use[T]g to verify that T is diagonalizable and T is an isomorphism.Important warnings about the solution: One MUST stick tothe notations given in the problem. Leave Pr,(A) as a productof linear factors, do NOT multiply the factors. One must usecofactor expansion while finding the characteristic polynomial in part(a). It is very straightforward to find the eigenvalues in part(a), hint is given to make sure that you find and use thecorrect eigenvalues. To be granted any points in each part youmust show your work, give explanations and write clear solutions. No

Given T:M2×2ℝ → M2×2(ℝ) is a linear operator given by, Tabcd=16a5b+11c11b+5c16d where M2×2(ℝ)…

Answered: Let T : M2x2(R) → M2x2(R) be the linear…

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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Question

Let T : M2×2(R) → M2×2(R) be the linear
operator given by
T
a b
c d =

16a 5b + 11c
11b + 5c 16d

,
where M2×2(R) denotes the vector space of 2 × 2-matrices over R.
Let B =
"
1 0
0 0
,

0 1
0 0
,

0 0
1 0
,

0 0
0 1
#
denote
the standard basis of M2×2(R).
(a) Find [T]B. Determine the characteristic polynomial of [T]B, i.e.
P[T]B
(λ). Find the eigenvalues of [T]B and determine the algebraic multiplicity of each eigenvalue.
(Hint: Eigenvalues are −6 and 16.)
(b) Determine a basis for each eigenspace E[T]B
(λ) of [T]B and determine the geometric multiplicity of each eigenvalue.
(c) Use parts a and b to determine the characteristic polynomial
PT (λ) of T, eigenvalues of T and a basis for each eigenspace
ET (λ) of T. Show that eigenvectors of T form a basis B0
for
M2×2(R).
(d) Find the representation [T]B0 of T relative to the basis B0
. Use
[T]B0 to verify that T is diagonalizable and T is an isomorphism.

Let T: M2x2(R) → M2x2(R) be the linear
operator given by
a b
16a
56 + 11c]
11b + 5c
16d
where M2x2(R) denotes the vector space of 2 x 2-matrices over R.
0 0
0 1
0 0
Let B =
denote
the standard basis of M2x2(R).
(a) Find [T]B. Determine the characteristic polynomial of [T]B, i.e.
Pr, (A). Find the eigenvalues of [T]B and determine the alge-
braic multiplicity of each eigenvalue.
(Hint: Eigenvalues are -6 and 16.)
(b) Determine a basis for each eigenspace ET, (A) of [T]B and de-
termine the geometric multiplicity of each eigenvalue.
(c) Use parts a and b to determine the characteristic polynomial
Pr (X) of T, eigenvalues of T and a basis for each eigenspace
ET(A) of T. Show that eigenvectors of T form a basis B' for
M2x2(R).
(d) Find the representation [T]g of T relative to the basis B'. Use
[T]g to verify that T is diagonalizable and T is an isomorphism.
Important warnings about the solution: One MUST stick to
the notations given in the problem. Leave Pr,(A) as a product
of linear factors, do NOT multiply the factors. One must use
cofactor expansion while finding the characteristic polynomial in part
(a). It is very straightforward to find the eigenvalues in part
(a), hint is given to make sure that you find and use the
correct eigenvalues. To be granted any points in each part you
must show your work, give explanations and write clear solutions. No

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