Problem 3 Let T: M2x2(R) → M2x2(R) be the linear operator defined as a b За - 2с + 4d T с d -2a + 6c + 2d 4a + 2c + 3d [[::}[::||:) 0 1 0 0 Let B = denote the standard basis of M2x2(R). 0 0 0 0 1 0 0 1 a) Find [T]B. Determine the characteristic polynomial of [T)B, i.e. PT (A). Find the eigenval- ues of [T]B and determine the algebraic multiplicity of each eigenvalue. (Hint: Eigenvalues are -2, 0 and 7.) b) Determine a basis for each eigenspace ErT, (A) of (T]B and determine the geometric multiplic- ity of each eigenvalue. Find it in increasing order of the eigenvalues.
Problem 3 Let T: M2x2(R) → M2x2(R) be the linear operator defined as a b За - 2с + 4d T с d -2a + 6c + 2d 4a + 2c + 3d [[::}[::||:) 0 1 0 0 Let B = denote the standard basis of M2x2(R). 0 0 0 0 1 0 0 1 a) Find [T]B. Determine the characteristic polynomial of [T)B, i.e. PT (A). Find the eigenval- ues of [T]B and determine the algebraic multiplicity of each eigenvalue. (Hint: Eigenvalues are -2, 0 and 7.) b) Determine a basis for each eigenspace ErT, (A) of (T]B and determine the geometric multiplic- ity of each eigenvalue. Find it in increasing order of the eigenvalues.
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 b
![Problem 3
Let T: Max2(R) → M2x2(R) be the linear operator defined as
a b
T
с d
За - 2с + 4d
-2a + 6c + 2d 4a + 2c + 3d
1 0
0 1
0 0
0 0
Let B =
denote the standard basis of M2x2(R).
0 0
0 0
1 0
0 1
a) Find [T]B. Determine the characteristic polynomial of [TB, i.e. PT(A). Find the eigenval-
ues of [T)B and determine the algebraic multiplicity of each eigenvalue.
(Hint: Eigenvalues are -2, 0 and 7.)
b) Determine a basis for each eigenspace ET, (A) of (T]B and determine the geometric multiplic-
ity of each eigenvalue. Find it in increasing order of the eigenvalues.
c) Show that eigenvectors of [T)B exhibit an orthonormal basis of R4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26a77bed-bbca-410f-a4fd-d651a789a370%2Ff5930a42-b616-45cd-bc9d-02216e6ec441%2Fb8kfzqb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 3
Let T: Max2(R) → M2x2(R) be the linear operator defined as
a b
T
с d
За - 2с + 4d
-2a + 6c + 2d 4a + 2c + 3d
1 0
0 1
0 0
0 0
Let B =
denote the standard basis of M2x2(R).
0 0
0 0
1 0
0 1
a) Find [T]B. Determine the characteristic polynomial of [TB, i.e. PT(A). Find the eigenval-
ues of [T)B and determine the algebraic multiplicity of each eigenvalue.
(Hint: Eigenvalues are -2, 0 and 7.)
b) Determine a basis for each eigenspace ET, (A) of (T]B and determine the geometric multiplic-
ity of each eigenvalue. Find it in increasing order of the eigenvalues.
c) Show that eigenvectors of [T)B exhibit an orthonormal basis of R4.
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