Let T : M2x2(R) → M2x2(R) be the linear operator given by 56 + 11c] 16d 16a T 116 + 5c where M2x2(R) denotes the vector space of 2 x 2-matrices over R. 0 1 Let B = denote the standard basis of M2x2(R). (a) Find [T]B. Determine the characteristic polynomial of [T]B, i.e. Pr»(1). Find the eigenvalues of [T], and determine the alge- braic multiplicity of each eigenvalue. (Hint: Eigenvalues are -6 and 16.) (b) Determine a basis for each eigenspace E, (A) of [T]B and de- termine the geometric multiplicity of each eigenvalue.
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
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda951066-7617-4533-aaa2-dd62371161e9%2Fe39f55bd-8a27-465d-9c27-404ba4e553f5%2Fnhg5a_processed.png&w=3840&q=75)
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