) Find the representation [T]B of T relative to the basis B'. Use [T]B to verify that T is diagonaliz- able, injective and surjective.
) Find the representation [T]B of T relative to the basis B'. Use [T]B to verify that T is diagonaliz- able, injective and surjective.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 29EQ
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please send handwritten solution for part iv Q2
![(ii) Determine a basis for each eigenspace ET,(A) of
[T]B and determine the geometric multiplicity of
each eigenvalue.
Problem 1:
Let
u1 = (m, 14, 7), u2 = (0, 1, –1) and uz = (2, –1, m)
(iii) Use parts i and ii to determine the characteristic
polynomial Pr(A) of T, eigenvalues of T and a
basis for each eigenspace Er(A) of T. Show that
eigenvectors of T form a basis B' for M2x2(R).
be vectors in R³.
(a) Find for which values of m, S = {u1, u2, uz} forms
a basis for R3.
(b) Let W = Span S =< u1; U2, Uz >. Determine a
basis and find dim(W) for each value of m e R.
(iv) Find the representation [T|B of T relative to the
basis B'. Use [T\B to verify that T is diagonaliz-
able, injective and surjective.
Important warning about Problem 1:
One must use cofactor expansion if one makes use of
the determinant.
Problem 2:
Let T : M2x2(R) → M2x2(R) be the linear operator
defined as
20a
186 + 2c
T
2b+ 18c
20d
0 1
0 0
denote the standard basis of M2x2(R).
0 0
1 0
Let B =
(i) Find [TB. Determine the characteristic polyno-
mial of [T]B, i.e. PT,(A). Find the eigenvalues
of [T]B and determine the algebraic multiplicity of
each eigenvalue.
(Hint: Eigenvalues are 16 and 20.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6b572ef-c150-472a-b31d-8f512ee37557%2F3f311412-176e-4e48-8fc4-3d7786d7f38f%2F6n35n4a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(ii) Determine a basis for each eigenspace ET,(A) of
[T]B and determine the geometric multiplicity of
each eigenvalue.
Problem 1:
Let
u1 = (m, 14, 7), u2 = (0, 1, –1) and uz = (2, –1, m)
(iii) Use parts i and ii to determine the characteristic
polynomial Pr(A) of T, eigenvalues of T and a
basis for each eigenspace Er(A) of T. Show that
eigenvectors of T form a basis B' for M2x2(R).
be vectors in R³.
(a) Find for which values of m, S = {u1, u2, uz} forms
a basis for R3.
(b) Let W = Span S =< u1; U2, Uz >. Determine a
basis and find dim(W) for each value of m e R.
(iv) Find the representation [T|B of T relative to the
basis B'. Use [T\B to verify that T is diagonaliz-
able, injective and surjective.
Important warning about Problem 1:
One must use cofactor expansion if one makes use of
the determinant.
Problem 2:
Let T : M2x2(R) → M2x2(R) be the linear operator
defined as
20a
186 + 2c
T
2b+ 18c
20d
0 1
0 0
denote the standard basis of M2x2(R).
0 0
1 0
Let B =
(i) Find [TB. Determine the characteristic polyno-
mial of [T]B, i.e. PT,(A). Find the eigenvalues
of [T]B and determine the algebraic multiplicity of
each eigenvalue.
(Hint: Eigenvalues are 16 and 20.)
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