Winter 2016
Math 325
Problems 2
01/20/2016
2 1 0
1) Let A = 0 2 2 . Compute A1 using the Cayley-Hamilton theorem.
1 0 1
2) Let A =
2 9
1 8
!
and A : C2 C2 , v 7 A v, the corresponding linear map.
(i) Show that A has only one eigenvalue and that A is not d
Dimensions of Generalized Eigenspaces
Generalized Eigenspaces of Jordan Matrices
Linear Algebra III Lecture 9
Xi Chen 1
1 University
of Alberta
February 4, 2015
Xi Chen
Linear Algebra III Lecture 9
Dimensions of Generalized Eigenspaces
Generalized Eigensp
Matrix Representations of Linear Transformations
Linear Algebra III Lecture 2
Xi Chen 1
1 University
of Alberta
January 9, 2015
Xi Chen
Linear Algebra III Lecture 2
Matrix Representations of Linear Transformations
Outline
1
Matrix Representations of Linea
Jordan Canonical Form For 2 2 Matrices
Linear Algebra III Lecture 7
Xi Chen 1
1 University
of Alberta
January 30, 2015
Xi Chen
Linear Algebra III Lecture 7
Jordan Canonical Form For 2 2 Matrices
Outline
1
Jordan Canonical Form For 2 2 Matrices
Xi Chen
Lin
Solutions for Math 325 Assignment #3
1
(1) For each of the following square matrices A, find an invertible
matrix P such that P 1 AP is upper-trangular.
3 1 0
4 1
a)
b) 1 1 0
9 2
1 0 2
Solution. a) Its characteristic polynomial is (x 1)2 and its
eigenspac
Minimal Polynomial
Linear Algebra III Lecture 11
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1 University
of Alberta
February 13, 2015
Xi Chen
Linear Algebra III Lecture 11
Minimal Polynomial
Outline
1
Minimal Polynomial
Xi Chen
Linear Algebra III Lecture 11
Minimal Polynomial
Minimal Pol
Triangularization
Linear Algebra III Lecture 6
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1 University
of Alberta
January 26, 2015
Xi Chen
Linear Algebra III Lecture 6
Triangularization
Outline
1
Triangularization
Xi Chen
Linear Algebra III Lecture 6
Triangularization
Triangularization
T
Cayley-Hamilton Theorem
Linear Algebra III Lecture 16
Xi Chen 1
1 University
of Alberta
March 16, 2015
Xi Chen
Linear Algebra III Lecture 16
Cayley-Hamilton Theorem
Outline
1
Cayley-Hamilton Theorem
Xi Chen
Linear Algebra III Lecture 16
Cayley-Hamilton Th
Solutions for Math 325 Final
1
(1) Which of the following statements are true and which are false?
Justify your answer.
(a) For a linear endomorphism T : V V on a vector space V ,
if R(T ) = R(T 2 ), then R(T 2014 ) = R(T 2015 ), where R(T )
is the range
Existence of Jordan Canonical Form
Linear Algebra III Lecture 19
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1 University
of Alberta
April 1, 2015
Xi Chen
Linear Algebra III Lecture 19
Existence of Jordan Canonical Form
Outline
1
Existence of Jordan Canonical Form
Xi Chen
Linear Algebra I
Solutions for Math 325 Assignment #2
1
(1) Let P3 be the vector space of real polynomials in x of degree at
most 3 and let T : P3 P3 be the linear transformation
T (f (x) = xf 0 (x + 1).
(a) Find the matrix [T ]BB representing T under the basis
B = cfw_1,
Generalized Spectral Theorem
Linear Algebra III Lecture 12
Xi Chen 1
1 University
of Alberta
March 4, 2015
Xi Chen
Linear Algebra III Lecture 12
Generalized Spectral Theorem
Outline
1
Generalized Spectral Theorem
Xi Chen
Linear Algebra III Lecture 12
Gene
Solutions for Math 325 Assignment #7
1
(1) Let T : V V be a linear endomorphism on a vector space V .
Show that every one-dimensional T -invariant subspace of V is
spanned by an eigenvector of T .
Proof. Let W be a one-dimensional T -invariant subspace of
Generalized Eigenspace
Linear Algebra III Lecture 8
Xi Chen 1
1 University
of Alberta
February 2, 2015
Xi Chen
Linear Algebra III Lecture 8
Generalized Eigenspace
Outline
1
Generalized Eigenspace
Xi Chen
Linear Algebra III Lecture 8
Generalized Eigenspace
Similarity, Eigenvalue, Eigenvector and Characteristic Polynomial
Linear Algebra III Lecture 4
Xi Chen 1
1 University
of Alberta
January 16, 2015
Xi Chen
Linear Algebra III Lecture 4
Similarity, Eigenvalue, Eigenvector and Characteristic Polynomial
Outlin
Winter 2016
Math 325
Problems 1
01/13/2016
1 4
!
and A : R2 R2 , v 7 A v be the corresponding R-linear map.
3 1
!
!
2
1
Give the matrix of the linear map A with respect to the basis
,
of R2 .
1
3
!
2 2
2) Let A =
. What are the A-invariant subspaces in R2
Winter 2016
Math 325
Praxis Midterm Exam
02/03/2016
1) Let
1 1
A =
1
!
,
3
and A : C2 C2 , v 7 A v be the corresponding C-linear map. Give a Jordan
basis for A .
(4 credits)
1 1 3
2) Let A = 0 1 1 . Use the Cayley-Hamilton Theorem to compute A1 .
1 0 1
(4
Winter 2016
Math325
Solutions to Problems 2
1) The characteristic polynomial of the matrix A is
PA (T ) = T 3 5T 2 + 8T 6 .
By the Cayley-Hamilton Theorem we have therefore
0 = PA (A) = A3 5 A2 + 8 A 6 I3 ,
and this is equivalent to the equation A
A1 =
1
Winter 2016
Math 325
Problems 5
03/02/2016
1) Compute a square root of the following 2 2-matrix:
!
3 1
A =
.
1 1
2) Compute exp(A) for
1 1
A =
2
4
!
.
(3) Give a Jordan basis and the Jordan
4 4-matrix:
3 1
1 1
A =
0 0
normal form for the following comp
Winter 2016
Math 325
Problems 3
01/27/2016
1) Determine the eigenvalues and the eigenspaces of the following two matrices
!
!
0 1
2 1
and
,
1 0
1 4
and if one of them is not diagonalizable give a Jordan basis for it.
3 2
0
2) Let A = 0 1 1 .
1 1
1
(i) Sho
Winter 2016
Math325
Solutions to Problems 3
0 1
1) Set A :=
1
0
!
and B :=
2 1
1 4
!
. We first compute the characteristic
polynomials:
PA (T ) = T 2 + 1 = (T ) (T + ) ,
where =
1 is square root of 1, and
PB (T ) = T 2 6T + 9 = (T 3)2 .
Therefore A is dia
Winter 2016
Math 325
Problems 4
02/24/2016
1) Compute the Jordan normal form of the following 3 3-matrix
3
1 2
A = 1 1 1 ,
2
1 1
and give a Jordan basis for A. (Hint: 1 is an eigenvalue of A.)
2) Compute the Jordan normal form of the following 3 3-matrix
Applications of Cayley-Hamilton Theorem
Linear Algebra III Lecture 17
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1 University
of Alberta
March 23, 2015
Xi Chen
Linear Algebra III Lecture 17
Applications of Cayley-Hamilton Theorem
Outline
1
Applications of Cayley-Hamilton Theorem
Xi Chen
Vector Space
Subspace
Basis and Dimension
Linear Transformation
Linear Algebra III Lecture 1
Xi Chen 1
1 University
of Alberta
January 7, 2015
Xi Chen
Linear Algebra III Lecture 1
Vector Space
Subspace
Basis and Dimension
Linear Transformation
Outline
1
V
Diagonalization
Linear Algebra III Lecture 5
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1 University
of Alberta
January 19, 2015
Xi Chen
Linear Algebra III Lecture 5
Diagonalization
Outline
1
Diagonalization
Xi Chen
Linear Algebra III Lecture 5
Diagonalization
Diagonalization of Linear E
Application to Linear Differential Equations
Linear Algebra III Lecture 18
Xi Chen 1
1 University
of Alberta
March 27, 2015
Xi Chen
Linear Algebra III Lecture 18
Application to Linear Differential Equations
Outline
1
Application to Linear Differential Equ
Invariant Subspaces
Linear Algebra III Lecture 13
Xi Chen 1
1 University
of Alberta
March 13, 2015
Xi Chen
Linear Algebra III Lecture 13
Invariant Subspaces
Outline
1
Invariant Subspaces
Xi Chen
Linear Algebra III Lecture 13
Invariant Subspaces
Invariant
Solutions for Math 325 Assignment #8
1
(1) Let T : V V be a linear endomorphism on a finite-dimensional
vector space. Suppose that
dim K(T I)k : k = 0, 1, . = cfw_0, 2, 2, .
dim K(T + I)k : k = 0, 1, . = cfw_0, 4, 6, 7, 7, .
dim K(T I)k = 0 for all 6=