Practice Problems: Final Exam
1. Find a basis for the subspace of R4 spanned by the following vectors:
 
 
 
 
 
1
4
1
2
5
 
 
 
 
 
1
5
4
5
7
 ,
 ,
 ,
 ,
 .
1
4
1
2
6
1
6
7
8
9
"
#
4 1
"
# "
#
"
#
1 1
1 0
1 2
as a linear combination of the matrices
,
, and
.
2. Express the matrix
0 1
1 1
0 1
3 4
3. 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.
4. Let T : R3 → R3 be the linear transformation that reflects across the plane x = z. Find the matrix
for T .
5. Find the general solution to each of the following equations.
(a) y (3) − 2y 00 + 5y 0 = 0
(b) y (3) − 5y 00 + 3y 0 + 9y = 0
(c) y (4) − 16y = 0
6. Let P2 be the vector space of all polynomials of the form p(x) = ax2 + bx + c, where a, b, c ∈ R,
and let S be the subspace of P2 consisting of all polynomials for which p(3) = 0. Find a basis
for S.


−5 −3 3


7. Find the eigenvalues of the matrix  3 1 −3 , and find a basis for each eigenspace.
−3 −3
1
8. Suppose that the vectors (1, 2, 5), (0, 1, 6), and (2, 7, c) span a plane in R3 . What is the value
of c?
9. Find the general solution to the following system of differential equations:
y10 = 4y1 ,
y20 = y22 ln(y1 ).
10. 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


1 2 2
1

11. The matrix 2 1 −2  acts as a reflection in R3 . Find a basis for the plane of reflection.
3
2 −2 1
12. Let S be the subspace of R4 spanned by the vectors (3, −5, 1, 0) and (−2, 4, 0, 1). Find a matrix
whose null space is S.
13. Find the solution to the following initial value problem:
y10 = 2y1 + y2 ,
y20 = 3y1 ,
y1 (0) = 9,
y2 (0) = 1.
14. Let A be a 5 × 4 matrix, and suppose that the null space of A is a line in R4 . What is the
dimension of the row space of A?