Linear Algebra I
Worksheet 7 Solutions
March 13, 2007
1. [12 pts] Consider the subset S = cfw_t2 + t, 2t2 + 1, t + 2 of vectors in P2 .
(a) Show directly that S is a basis.
We need to show that S span
Linear Algebra I
Worksheet 3 Solutions
February 6, 2007
1. [12 pts] Let L : R4 R4 be the linear transformation dened by
ab
a
b 0
L =
c c d .
0
d
(a) Show directly that L is linear.
We first show
Linear Algebra I
Worksheet 1 Solutions
January 23, 2007
1. [6 pts] Use Gaussian elimination to solve the following linear system:
x + y + 2z = 1
x
+z =2
2x + y + 3z = 1
Beginning with the correspondin
Linear Algebra I
Test 3 Solutions
April 19, 2007
1. Prove the Pythagorean theorem: If u and v are orthogonal, then
u+v
2
= u
2
+ v
2
.
Note that
u+v
2
= u + v, u + v = u, u + 2 u, v + v, v
The fact th
Linear Algebra I
Final Exam
Name:
December 13, 2004
OUID:
Instructions: No calculators allowed. For full credit be sure to show all of your work
and indicate your answer clearly. Unjustied answers wil
Linear Algebra I
Test 2 Solutions
March 15, 2007
1. Let L : P2 P2 be dened by L(at2 + bt + c) = (2a c)t2 + (a + b c)t + (c).
(a) Find the matrix representation for L with respect to the basis (on both
Linear Algebra I
Worksheet 5 Solutions
February 27, 2007
1. [9 pts] You are given the following vector and two ordered bases for R3 :
0
0
1
0
7
1
2
0 , 1 , 1
1 , 0 , 1
1
S=
T =
v=
1
2
3
0
3
Linear Algebra I
Worksheet 11 Solutions
1. Compute eA , where A =
May 3, 2007
1 1
2 4
We first compute eigenvalues, finding that
det
1
1
= (1 )(4 ) + 2 = 2 5 + 6 = ( 2)( 3).
2 4
So the eigenvalues ar
Linear Algebra I
Worksheet 10 Solutions
April 17, 2007
1. [3 pts] Suppose is an eigenvalue of the n n matrix A. Show that the set of
all eigenvectors associated with (along with the zero vector) is a
Linear Algebra I
Worksheet 8 Solutions
April 3, 2007
1. Let V be the vector space of continuous functions on [0, 1] and dene an inner
1
product by f, g = 0 f (t)g(t) dt.
(a) Find the cosine of the ang
Linear Algebra I
Worksheet 4 Solutions
February 13, 2007
1. [3 pts] Do the following vectors span R4 (justify your answer):
0
1
1
1
0 1 0 1
,
, ,
5
2 1 0
1
2
2
2
We want to know if for any a
Linear Algebra I
Test 1 Solutions
1. (a) Show that the set of vectors of the form
February 15, 2007
a
, where a is an integer, is not a
b
subspace of R2 .
This set is not closed under scalar multiplic