Discussion Points
K:
Linear Space over K:
Zero Vector Notation:
Examples of Linear Spaces:
What is a subspace of a linear space?
What is the trivial subspace?
What do all of the subspaces of 2 look like?
What do all of the subspaces of 3 look like?
Exampl
MATH 4378, LINEAR ALGEBRA II HW#10, due on Thursday. August 2
1. Let T : V V be a linear operator over a nite dimensional vector space
V . Let and be two eigenvalues of T .
(a) Prove that if = , then E E = cfw_0 where E and E are
eigenspaces of and respec
August 1, 2012
Homework 9
Math 4378
Question 2, (a):
11
1 3
A=
The characteristic polynomial of A is f (t) = (t 2)2 . Hence = 2 is the eigenvalue of A with
multiplicity 2. dim(K ) = 2.
(A I )v = 0
=
1 1
1 1
=
0
0
x + y
x + y
=
x
y
=
0
0
= x = y
Thus [1 1]
MATH 4378, LINEAR ALGEBRA II Key to Quiz#3
1. Consider R3 with the standard inner product. Let v = (1, 2, 1). Find an
orthonormal basis for cfw_v . (Hint: Find a basis rst, then use Gram-Schmidt
process).
Solution: By the denition,
cfw_v = cfw_(x, y, z
Math 4378, Linear Algebra II, Review for the First Exam.
Summer 2012, Dr. Min Ru, University of Houston
In this notes, T is always a linear operator on V , V is a always a vector space, A
is always an n n matrix, . . .
Part I: Questions
What are the dein
UH - Math 4378 - Dr. Heier - Spring 2011
Quiz 1 02/02/2011
Name:
1. (5 points) Let T : V V be an invertible linear operator. Prove that a scalar is an eigenvalue of T
if and only if 1 is an eigenvalue of T 1 .
2. (5 points) Let S : V V be a linear operato
MATH 4378, LINEAR ALGEBRA II Key to Quiz#1
1. Let V be a nite-dimensinal space and T : V V be a linear operator
on V . A subspace W V is T -invariant if and only if ll it out: For
w W , T (w) W .
2. Let T be the linear operator on R4 dened by T (a, b, c,
Jordan Canonical Form (The note written by Min Ru)
Key words. Jordan canonical form, canonical basis, eigen-space, generalized
eigenspace
I intended to make this notes to be a self-contained, i.e. all theorems will be
proved here.
1
Introduction
The goal
MATH 4378, LINEAR ALGEBRA II, Key to HW#10
1. Let T : V V be a linear operator over a nite dimensional vector space
V . Let and be two eigenvalues of T .
(a) Prove that if = , then E E = cfw_0 where E and E are
eigenspaces of and respectively.
(b) Use (a)
MATH 4378, LINEAR ALGEBRA II HW1
1.(1)
Proof. Assume there exists c1 and c2 such that c1 (1, 1) + c2 (1, 1) = (0, 0),
then c1 + c2 = 0 and c1 c2 = 0. By solving the system of two equations, we
have c1 = c2 = 0 and therefore (1, 1) and (1, 1) are linearly
MATH 4378, LINEAR ALGEBRA II Key to HW6
6.3 #3(b)
If is the standard ordered basis for C2 , then [T ] =
2
i
1i 0
.
So
[T ] =
2 1+i
i
0
.
Hence T (z1 , z2 ) = (2z1 + (1 + i)z2 , iz1 )
and T (3 i, 1 + 2i) = (5 + i, 1 3i).
6.3 #8
Proof. Let T 1 be the invers
1. Suppose
is a subspace of
. Prove
.
NOTE: Problems 2 and 3 below are the justifications for us being able to discuss the notion of the projection of
a vector onto a subspace of .
is a vector so that | |
2. Definition: A projection of a vec
Math 4378, Linear Algebra II, Key to Review and Mock-Second Exam.
Summer 2012, Dr. Min Ru, University of Houston
1. Let
A=
3
1 2i
1 + 2i 1
.
(a) Quote a theorem to see why you can nd an unitary matrix U such
that U 1 AU is diagonal.
(b) Find such unitary
MATH 4378, LINEAR ALGEBRA II Key to HW3
6.1 #2
x, y = x y = (2 + i, i) (2 + i, 2, 1 2i) = 8 + 5i.
,1
x=
x, x = x x = (2, 1 + i, i) (2, 1 i, i) = 7.
y=
y , y = y y = (2 i, 2, 1 + 2i) (2 + i, 2, 1 2i) = 14.
x + y = (4 + i, 3 + i, 1 + 3i), and
x + y = (4 + i
Math 4378, Linear Algebra II, Review and Mock-Second Exam.
Summer 2012, Dr. Min Ru, University of Houston
1
Review Problems
List the dening properties of an inner product.
Provide the denition of an orthogonal and an orthonormal basis.
How would you ex
MATH 4378, LINEAR ALGEBRA II Key to HW7
6.4 #2(c)
Let = cfw_(1, 0), (0, 1), and then T (1, 0) = (2, 1), T (0, 1) = (i, 2). The matrix
representation of T in is
2i
12
A=
and A =
21
i 2
.
Thus,
AA = A A =
5
2i + 2
2i + 2
5
but A = A , so T is normal but not
MATH 4378, Hey to LINEAR ALGEBRA II HW8
2.3 #17 Let V be a vector space. Determine all linear transforamations
T : V V such that T = T 2 . Hint: Note that x = T (x) + (x T (x) for
every x V , and show that V = cfw_y : T (y ) = y N (T ).
Proof. Recall the
MATH 4378, LINEAR ALGEBRA II HW#5, due on Tuesday, July 17
1. On R3 with the usual inner product. Use Gram-Schmidt to convert
x1 = (1, 2, 0), x2 = (3, 1, 1), x3 = (4, 3, 5) to an orthonormal basis.
2. (a) Let V be an inner product space and W be a nite di
MATH 4378, LINEAR ALGEBRA II HW#1, due on Wed. July 11.
The purpose of this set of homework is to review some ADVANCED
LINEAR ALGEBRA I material.
1. Let T be the linear operator on R2 dened by
T
a
b
3a b
a + 3b
=
,
and let = cfw_(1, 1), (1, 1) and = cfw_(
MATH 4378, LINEAR ALGEBRA II, Key to HW2
5.3 #13
We observe that the transition matrix A and the initial probability vector
P1975 of the Markov chain are
0.7 0.1 0
0.4
A = 0.3 0.7 0.1 and P1975 = 0.2 .
0 0.2 0.9
0.4
0.24
Thus, P1995 = A2 P1975 = 0.34 whic
Invariant Subspaces
Min Ru
1
T -invariant subspaces
The notion of a T -invariant subspace of V is rather stragightforwad, but the consideration of such spaces is one important ingredient in understanding the simplest
form of a linear transformation (it wi
August 8, 2012
Key to Homework 11
Math 4378
Question 1(a):
Suppose a0 , a1 , a2 , a3 , ., ar1 belongs to F such that
a0 x + a1 (T I )x + a2 (T I )2 x + . + ar1 (T I )r1 x = 0
Because (T I )r x = 0, if we apply (T I )r1 to the both sides of the above equat
Practice Exam 3
Advanced Linear Algebra II - G. Guidoboni - Spring 2009
First and Last Name: .
Show all your work in answering the following questions. Simply writing the
answer is not acceptable.
1. Consider R2 with the standard inner product. Let T be t
Practice Exam 2
Advanced Linear Algebra II - G. Guidoboni - Spring 2009
First and Last Name: .
Show all your work in answering the following questions. Simply writing the
answer is not acceptable.
1. Consider R4 with the standard inner product. Let W be t
Practice Exam 1
Advanced Linear Algebra II - G. Guidoboni - Spring 2009
First and Last Name: .
Show all your work in answering the following questions. Simply writing the
answer is not acceptable.
1. Let T be a linear operator on R2 dened by
T (x1 , x2 )