Linear Algebra I
Test 1 Solutions
June 15, 2007
1. (a) Suppose W is a subspace of R5 . What can you say (if anything) about the
dimension of W ?
We know that dim(W ) 5 . This is because a basis for W consists
of vectors that are linearly independent in R5
Math 3333 homework
9. (2/24) 2.2 # 23 As asked in the book, nd the condition on a, b, and c that allows
solutions. Then, analyze the solutions as we did in class: nd all solutions for a xed
a, b, and c, and write them as a particular solution plus a solut
Math 3333 homework
22. (4/9) 5.3 # 11, 12 (where the length of x is dened by x = (x, x)1/2 ), 15, 16 (expand
the left-hand side of the formula using the fact that x 2 = (x, x), 19 (rst assume
that the equality holds and check that (u, v) = 0, then assume
Mathematics 3333-002
Name (please print)
Examination III Form B
April 28, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation
Mathematics 3333-002
Name (please print)
Examination III Form A
April 28, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation
Mathematics 3333-002
Name (please print)
Examination II Form A
March 24, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation,
Mathematics 3333-002
Name (please print)
Examination II Form B
March 24, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation,
Name (please print)
Mathematics 3333-002
Examination II Form A
March 24, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation,
Mathematics 3333-002
Name (please print)
Examination I Form B
February 19, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculatio
Mathematics 3333-002
Name (please print)
Examination III Form A
April 28, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation
Name (please print)
Mathematics 3333-002
Examination II Form B
March 24, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculation,
Mathematics 3333-002
Name (please print)
Examination I Form A
February 19, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculatio
Mathematics 3333-002
Name (please print)
Final Examination Form A
May 11, 2010
Instructions: Give concise answers, but clearly indicate your reasoning.
I.
(4)
Let V be an inner product space, that is, a vector space V equipped with an inner product denote
Mathematics 3333-002
Name (please print)
Examination I Form B
February 19, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculatio
Linear Algebra I
Test 2 Solutions
June 29, 2007
0
0
1
3
2
1 , 1 , 0 is a basis
1. Suppose L : R R is a linear map, S =
1
1
1
3
2
1 32
for R3 , T =
,
is a basis for R2 , and A =
is the matrix
4
1
1 0 1
2
1 .
representation for L with respect to S and
Linear Algebra I
Worksheet 2 Solutions
June 13, 2007
a
b
1. (a) [3 pts] Show that the set of all vectors of the form , where c = a + 2b and
c
d
4
d = a 3b , is a subspace of R .
We went over the direct way of doing this today in class.
a shortcut way:
Thi
Linear Algebra I
Worksheet 3 Solutions
June 22, 2007
a
bd
b a + b + c
1. Consider the linear map L : R4 R4 given by L =
c a + 4d .
d
3b + c d
(a) [3 pts] Find the standard matrix representation A for L .
0
1
A=
1
0
1
1
0
3
0 1
1 0
0 4
1 1
(b) [4 pts
Linear Algebra I
Worksheet 4 Solutions
June 27, 2007
1. Consider the following bases for R3 :
0
1
1
T = 1 , 1 , 0
0
1
0
0
0
1
T = 1 , 1 , 0
1
1
1
(a) Find the transition matrix QT T .
To get the columns of Q , we need to find the T -coordinates of
th
Linear Algebra I
Worksheet 5 Solutions
July 6, 2007
1. Suppose V is the set of all positive real numbers, and dene operations and
as follows:
if u is the positive real number a , and v is the positive real number b , then
dene u v to be ab (so is actuall
Linear Algebra I
Worksheet 7 Solutions
1. Compute eA , where A =
July 23, 2007
11
.
2 4
First we find the eigenvalues of A :
det(A I ) = ( 2)( 3),
so eigenvalues are = 2 and = 3 .
It follows that
Next we solve the system:
1 0 | e3 e2
3 1 | e3
.
2;
0 1 | 3
Linear Algebra I
Worksheet 8 Solutions
Not Due
a
c
1. Consider the inner product space R2 with
,
= ac ad bc + 3bd.
b
d
2
(a) Find the length of
.
3
Using the formula
v=
v, v , we have
2
= 4 6 6 + 27 = 19.
3
1
1
and
.
0
1
v, w
Using the formula = cos1
vw
(
Math 3333 homework
1. (due 2/1) 1.1 # 2, 7, 8, 11, 12, 17
2. (2/1) Be able to do any of 1.2 # 1 and 4-12. Turn in 4 and 5 (use the method of
3 0
1 0
elimination using the operations of type I, II, and III), 9, 10 (solve
=a
+
0 2
0 1
1 0
b
), 11, 12
0 0
3.
Mathematics 3333-002
Name (please print)
Examination I Form A
February 19, 2010
Instructions: Give concise answers, but clearly indicate your reasoning. Most of the problems have rather short
answers, so if you nd yourself involved in a lengthy calculatio