Math 203–001, Final Exam.
Name:
200 pts, 180 minutes
Dec. 14, 2009
Page 1
Read each problem carefully. No calculators, notes, books, or any outside materials.
Unless otherwise indicated, supporting work will be required on every problem;
one-word answers or answers which simply restate the question will receive no credit.
  
   

−1
3
5 
 2
1 (4 pts). Is  0  ,  2  ,  4  ,  5  linearly independent in IR3 ?


3
0  2
−6

 a

2 (8 pts). Let H be the set  b  ∈ IR3 : 2a − 3b = c . Is H a subspace of IR3 ?


c
3a (8 pts). Let D denote the set {t2 , t(t − 1), (t − 1)(t − 2), t(t − 2)} of four polynomials in
IP2 . Does D span IP2 ?
3b (4 pts). Is D a basis for IP2 ?
4 (4 pts). If a set of five vectors {v1 , v2 , . . . , v5 } spans the vector space V , what can you
conclude about the dimension of V ?
p(0)
2
5. Let T be the transformation from IP2 into IR is given by the rule T p(t) =
.
p(1)
a (3 pts). Find T (t2 + 1).
b (8 pts). Prove that the transformation T is linear.
c (8 pts). Let B denote the basis {1, t, t2 } for IP2 and let C denote the standard basis
{e1 , e2 } for IR2 . Find the matrix of T relative to the bases B and C.
d (6 pts). Find a polynomial p(t) (other than 0) which is in the kernel of T . Include in
your answer an explanation
ofwhat it means
to belong to the kernel
for p(t) of T . 1
3
−2
−1
1 −1
6. Suppose
X −1 =
Y −1 =
and
Z −1 =
.
0 2
0
3
0 1
a (3 pts). Find (X T )−1 .
b (6 pts). Find det Y .
c (6 pts). Find (XY Z)−1 .
 




x1
1
2 0 1 0
 x2 
 0 
 3 2 0 3
7 (12 pts). Let A = 
 and x =  .
 and b = 
x3
0
α 1 1 1
x4
−1
β 1 1 1
If det A = 4 and Ax = b, find x1 .
8a (12 pts). Solve for x. Write your solution in parametric vector form.
x1
2x1
−10x1
− 2x2
− 4x2
+ 20x2
+ x3
+ x3
− 3x3
+ 5x4
+ 7x4
− 29x4
8b (3 pts). Find a basis for the nulspace of the matrix

=
6
=
7
= −25
1
 2
−10
−2
−4
20

1
5
1
7 
−3 −29
203–001 (Kunkle) Final Exam 12/14/09
No calculators
8c (3 pts). What is the rank of the matrix in 8b?
(No work required. A correct answer is sufficient
 for full credit.)

1 1 −1
9 (12 pts). Find the inverse of the matrix  1 2 0 
0 1 0
10 (8 pts). Suppose that A and B are row equivalent, where

a
A = d
g

b c
e f
h i
and

1 −1
B = 0 0
0 0

0
1.
1
a. Find a basis for Col A
b. Find a basis for Col B
c. Find a basis for Row A
d. Find a basis for Row B
  


−1 
1
−2



1  1 
 −2 
11a (6 pts). Find the orthogonal projection of 
 .
 onto span   , 
1 
3


 0
0
0
2
11b (14 pts). Produce an orthogonal set of vectors by applying Gram-Schmidt to
  

 
 
−3 
−2
−1
1



 1   1   −2   5 

,
,
 ,
1 
3
1


 0
3
2
0
0
4
11c (4 pts). Is the set you produced in 11b a basis
for IR
 ?
1
12 (4 pts). Find a unit vector in the direction of  2 .
−1


3 1 0
13 (10 pts). Is the matrix  0 3 0  diagonalizable? Explain.
0 0 1
14a (10 pts). Find the least-squares solution to Cx = f where


1
C =  −1
1

2
1
0
and

1
f = 1 
−5

p. 2
203–001 (Kunkle) Final Exam 12/14/09
No calculators
p. 3
14b (4 pts). Find the orthogonal projection of f onto the span of the columns of C.
14c (4 pts). Find the distance from f to the span of the columns of C.
15 (12 pts). Suppose Q is a linear map from IR5 into IR4 .
a. Could Q be one-to-one? If so, provide an example. If not, explain.
b. Could Q be onto? If so, provide an example. If not, explain.
 
 
1
1



16 (6 pts). If G is a 3×3 matrix and G 0 = G 1 , find an eigenvalue and eigenvector
1
0
of G.
17 (8 pts). Find all eigenvalues of
−4
−6
3
.
5