Practice Problems: Final Exam
1. Consider the following vectors:
 
 
1
2
 
 
v1 =  2 , v2 =  5 ,
1
3


4
 
v3 =  11 ,
6
and
 
3
 
w =  8 .
7
Express the vector w as a linear combination of the vectors v1 , v2 , and v3 .
2. Let S be the subspace of R4 spanned by the following three vectors:
   
 
1
2
4
   
 
2 3
5
 ,  , and  .
4 3
1
2
2
2
(a) Find a basis for S.
(b) What is the dimension of S?
3. Find the general solution to the following differential equation:
y (3) + 3y 00 + y 0 − 5y = 0.
"
4. Let A =
2
1
−2
0
#
.
(a) Find the eigenvalues of A.
(b) Find one eigenvector for each eigenvalue.
5. Find the value of c for which the following three vectors are linearly dependent:
 
 
 
3
2
6
 
 
 
1
5
c
 ,
 ,
 .
4
5
1
2
1
5
6. Find the eigenvalues of the following matrix:

2 8 0 7 3

 0 3 0

 6 2 9


 0 4 0
1 3 0


0 0 

8 6 


7 0 
1 4
7. Solve the following initial value problem:
y 00 + 6y 0 + 9y = 0,
y(0) = 4,
y 0 (0) = 5.
8. Let S be the subspace of R5 defined by the following equations:
x1 + 3x2 + 2x3 + 4x4 + x5 = 0
2x1 + 6x2 + 3x3 + 5x4 + 5x5 = 0
3x1 + 9x2 + 4x3 + 6x4 + 9x5 = 0
Find a basis for S.
9. Find all values of x for which the following three vectors are linearly dependent:
 
 
 
2
0
x
 
 
 
 x ,
 3 ,
 5 .
3
1
3

3

 0
10. Let A = 
 0

0

2 −8
5
2
4
0
3
0
0

3 
.
1 

3
(a) Find the eigenvalues of A.
(b) Find a basis for each eigenspace.
11. Let S be the subspace of R4 defined by the equation
x1 + 2x2 − 4x3 + 3x4 = 0,
and let T be the subspace of R4 defined by the equation
x1 + 6x3 + 7x4 = 0.
(a) Find a basis for S.
(b) Find a basis for T .
(c) Find a basis for the intersection of S and T .
12.
(a) Find the general solution to the following differential equation:
xy 00 − 2y 0 = 0.
(b) Find the general solution to the following differential equation:
xy 00 − 2y 0 = −2.
13. Find a 2 × 2 matrix A that satisfies the following conditions:
" #
1
•
is an eigenvector for A, with eigenvalue 2, and
1
" #
2
•
is an eigenvector for A, with eigenvalue 5.
1