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 spans P2 and is linearly independent.
To show it spans, let
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 that L(u + v) = L(u) + L(v). To this end we have
a
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 corresponding augmented matrix,
the following elementary row reduct
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 that u and v are orthogonal means that u, v = 0, so
that
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 will not receive any credit. Each
problem is worth a total
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 sides)
S = cfw_t + 1, t2 + t, t2 + t + 1.
We first plu
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
0
9
(a) Find the coordinate vectors [v]T and [v]S direc
Linear Algebra I
Worksheet 6 Solutions
March 6, 2007
1. Consider the following bases for R2 and R3 :
1
1
,
2
1
0
1
1
T = 1 , 1 , 0
0
1
0
S=
1
1
,
1
0
0
0
1
T = 1 , 1 , 0
1
1
1
S =
and note that the transition matrices are given by
PSS =
2/3 1/3
1
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 are = 2 and = 3.
e2 = 2a1 + a0
e3 = 3a1 + a0
;
We now sol
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 subspace of R n .
(This is called the eigenspace associ
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 angle between t2 and t.
cos =
t2 , t
t2
1 3
t
0
1/2
=
t
1
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, b, c, d, there is always a solution
to the system
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 multiplication. For example,
1
=
, which does not have an intege