Math 121B Linear Algebra
Quiz 1, April 11
Name:
Problem I: True or False. Circle the answer please.
T/F
Let A be a square matrix. If B is the matrix given by multiplying each entry of
A by 2, then detB = 2 detA.
False, in general we have det (IA) = n A fo
121B: Quiz 2
Non-graphical calculators allowed. No books or notes allowed. Each of the 2 parts
is worth the same.
Provide computations and or explanations (unless stated otherwise).
Name:
Student ID:
Exercise 1: Let V = P (R) (vector space of polynomials
121B: Quiz 4
Non-graphical calculators allowed. No books or notes allowed.
Provide computations and or explanations (unless stated otherwise).
Name:
Student ID:
Exercise 1 (7 pts):
Let
2
A= 1
1
1
2
1
1
1 .
2
Find an orthogonal matrix P and a diagonal matr
121B: Quiz 6
Non-graphical calculators allowed. No books or notes allowed.
Provide computations and or explanations (unless stated otherwise).
Name:
Student ID:
Exercise 1 (10 = 3.33333 + 3.33333 + 3.33334 pts):
Consider the real 2 2 matrices A and B give
121B: Quiz 1
Non-graphical calculators allowed. No books or notes allowed. Each of the 4 parts
is worth the same.
Provide computations and or explanations.
Name:
Student ID:
Exercise 1: Let T : R4 R4 be the linear operator defined by
T (a, b, c, d) = (a +
121B: Quiz 5
Non-graphical calculators allowed. No books or notes allowed.
Provide computations and or explanations (unless stated otherwise).
Name:
Student ID:
Exercise 1 (10 pts):
Let
A=
1
2
2
1
.
Pk
Find the spectral decomposition of A, that is, write
121B: Quiz 3
Non-graphical calculators allowed. No books or notes allowed. Each of the 2 parts
is worth the same.
Provide computations and or explanations (unless stated otherwise).
Name:
Student ID:
Exercise 1:
Find the minimal solution to the following
MATH 121B, SPRING 2011
PRACTICE PROBLEMS FOR THE FINAL
Problem 1. Suppose that T : V V is an invertible diagonalizable
linear operator. Show that T 1 is also diagonalizable.
Problem 2. Let
1 3 0
A = 0 2 1
0 0 0
Show that A is diagonalizable over R and fi
Probe
11
(a)
suffices
It
span cfw_
W=
distinct
ti
According
Teves
Therefore
Rink
all
,
( AZ )
w
T
by
functions
.
(b) By
.
a
,x'
be
it
,
to
it
aoto
,K
.
,
,
aoI=
I
At
th
=
=
"
only
.
,
and
EF
ai
Then
=W
WCS
show
to
want
Tv
,
.
set
generating
the
contains
i
121B: Quiz 8
Non-graphical calculators allowed. No books or notes allowed.
Provide computations and or explanations (unless stated otherwise).
Name:
Student ID:
Exercise 1 (5 points) Give a representative in Jordan normal form of all similarity
classes of
121B: Quiz 7
Non-graphical calculators allowed. No books or notes allowed.
Provide computations and or explanations (unless stated otherwise).
Name:
Student ID:
Exercise 1 (10 = 2 + 1 + 3 + 4 pts):
Consider the 3 3 real matrix A given by
3 1 2
A = 1 1 2 .
Math 121B Linear Algebra
Quiz 2, April 18
Name:
Problem I: Let A Mn (C) be a matrix whose characteristic polynomial is (2t)4 (3t)3 .
Suppose that dim E2 = 2 and dim E3 = 1. Write down the possible Jordan canonical forms
that A can take (up to reordering t
Math 121B Linear Algebra
Quiz 3, April 25
Name:
Problem I: Let A be a complex matrix whose characteristic polynomial is (t 3)10 .
Assume that we know dim E3 = 4 and dim N (A 3I )2 = 7. What are the possibilities of
the Jordan canonical form of A?
Solution
Math 121B Linear Algebra
Quiz 4, May 14
Name:
Problem I: Let cfw_v1 , v2 be an orthonormal basis for an inner product on R2 . What is
|3v1 + 4v2 |?
By orthonormality, vi , vj = ij . By expanding the square of the norm, we have
|3v1 + 4v2 |2 = 3v1 + 4v2 ,
Math 121B
Linear Algebra II
Name: .
Liang Xiao
Spring 2013
Student ID: .
MIDTERM
Friday, May 3, 2013
You have 50 minutes to complete the exam. Do all work on this exam, i.e., on the page
of the respective assignment. Indicate clearly, when you continue yo
Math 121B Linear Algebra
Quiz 5, May 21
Name:
Problem
I: Consider the standard inner product on V = R3 . Give an orthonormal
1
basis of Span(1) .
0
1
0
2
1 , 0
One possible basis is cfw_
2
1
0
0
1
2
Problem II: Consider points
,
,
. Use the least squar