(ii) Det. [T]. Problem 1: Let eac u = (m, 5, 2), u2 = (0,1, –1) and uz = (8, –1, m) (iii) Use pol basi be vectors in R°. (a) Find for which values of m, S = {u1, u2, U3} forms a basis for R. %3D eige
(ii) Det. [T]. Problem 1: Let eac u = (m, 5, 2), u2 = (0,1, –1) and uz = (8, –1, m) (iii) Use pol basi be vectors in R°. (a) Find for which values of m, S = {u1, u2, U3} forms a basis for R. %3D eige
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 58EQ
Related questions
Question
handwritten solution for part a Q 1
![(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, 5, 2), u2 = (0, 1, –1) and uz = (8, –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 R.
(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.
Important warnings about Problem 2:
One MUST stick to the notations given in the problem.
Leave P,(A) as a product of linear factors, do
NOT multiply the factors.
One must use cofactor expansion while finding the
characteristic polynomial in part (i).
It is very straightforward to find the eigenvalues
in part (i), hint is given to make sure that you
find and use the correct eigenvalues in the rest
of the problem.
To be granted any points in any part of the exam you
must show your work, give explanations and write clear
solutions. No work=No Credit.
Problem 2:
Let T : M2x2(R) → M2x2(R) be the linear operator
defined as
156 + 4c]
19d
19a
4b+15c
1 0
0 0
0 1
0 0
0 0
1 0
Let B =
denote the standard basis of Mx2(R).
(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 11 and 19.)
Important warning about the submission:
Problem 1 should be solved on the first page.
Problem 2, part i, should be solved on the second page.
Problem 2, part ii, should be solved on the third page.
Problem 2, parts iii and iv should be solved on the
fourth page. An extra page can be used if absolutely
necessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6b572ef-c150-472a-b31d-8f512ee37557%2F29f1b6eb-dc57-4296-b2b9-c5b9ed5efe22%2Fzietod_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, 5, 2), u2 = (0, 1, –1) and uz = (8, –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 R.
(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.
Important warnings about Problem 2:
One MUST stick to the notations given in the problem.
Leave P,(A) as a product of linear factors, do
NOT multiply the factors.
One must use cofactor expansion while finding the
characteristic polynomial in part (i).
It is very straightforward to find the eigenvalues
in part (i), hint is given to make sure that you
find and use the correct eigenvalues in the rest
of the problem.
To be granted any points in any part of the exam you
must show your work, give explanations and write clear
solutions. No work=No Credit.
Problem 2:
Let T : M2x2(R) → M2x2(R) be the linear operator
defined as
156 + 4c]
19d
19a
4b+15c
1 0
0 0
0 1
0 0
0 0
1 0
Let B =
denote the standard basis of Mx2(R).
(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 11 and 19.)
Important warning about the submission:
Problem 1 should be solved on the first page.
Problem 2, part i, should be solved on the second page.
Problem 2, part ii, should be solved on the third page.
Problem 2, parts iii and iv should be solved on the
fourth page. An extra page can be used if absolutely
necessary.
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