Math 237
Carter
Final
Summer 2013
General Instructions. Do not write on this test sheet. Do all your work inside your
blue book. Write neat, complete solutions to each of the problems below. When asked for a
specific calculation, check your results! If your result seems incorrect and you do not have
time to find an arithmetic mistake, then indicate that you know there is a mistake.
Stewed tomatoes and okra is an easy dish to prepare. Sauté vidalia onions in olive oil
until clear. Add fresh (homegrown) chopped tomatoes, and sliced okra. Add salt and sugar
to taste.
1. Fill in the blank (5 points each):
(a) If the homogeneous equation
x1~v1 + x2~v2 + . . . xk~vk = 0
has only the trivial solution x1 = x2 = . . . = xk = 0, then the set {~v1 , ~v2 , . . . , ~vk } ⊂
V is said to be
.
of a matrix A ∈ M (m, n) is the set
(b) The
~ : AX
~ = ~0) }.
{X
(c) If the vector space V has a basis {~v1 , ~v2 , . . . , ~vk }, then
of V
is k, that is, the number of elements in a linearly independent set that spans V .
F
for a linear operator Rn ←− Rn is a vector ~v ∈ Rn such
(d) An
that there is a λ ∈ R with
A~v = λ~v .
The number λ is said to be the associated
.
F
(e) Let U ←− V denote a linear map from an n-dimensional vector space V to an mdimensional vector space U and let A ∈ M (n, m) denote any matrix representing
F.
(f) The
states that
rank(F ) + dim(ker(F )) = n.
2. (5 points) Consider a matrix A ∈ M (5, 10). Suppose that the rank of A is 3. What
is the dimension of the null space? Is the linear map that is represented by A with
respect to the standard basis surjective (onto)?
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3. (10 points) Give a parametric equation for the line perpendicular to the plane x − 2y +
6z = 18 that contains the point P (2, 3, −7).
4. (10 points) Write the general solution set to the equation
2x − y + 3z + 7w − 2u = 42
in vector form.
5. The reduced row echelon form of the matrix


1
4
5
1 2



 1
3
4
−1
1

A=

 1
8
9
−1
0


−1 −1 −2 5 1
is

1 0 1 0

 0 1 1 0
A =
 0 0 0 1

0 0 0 0
0
11
5
− 15
3
5






0
(a) (5 points) Give a basis for the row space of A.
(b) (5 points) Give a basis for the column space of A.
(c) (5 points) Give a basis for the null space of A.
6. Compute the matrix product of the following matrices (10 points each).
(a)
"
5 1
1 5
# "
·
1
1
#
1 −1
(b)

 
 

1
0 0
1 1 0
1 1 −2 −2 1

 
 

 0 −1 0  ·  0 1 0  ·  0 −1 2

4 1 

 
 

−2 0 1
0 1 1
2 1 −3 −2 1
7. Solve the system of equations (10 points):
x
+ y − 2z − 2w = 1
− y + 2z + 4w = 1
2x + y − 3z − 2w = 1
2
8. (10 points) Write the matrix
" √
2
B=
e
π
#
17
as a product of elementary matrices. Recall that a matrix is an elementary matrix
if it is obtained from the identity matrix by means of an elementary row operation.
F
9. (10 points) Let R2 ←−R4 denote the linear mapping that is given by

x

#
  "
 y 
3x
−
2y
+
0z
+
2t

F
 z  = 0x + 2y + 3z + 2t .
 
t
Find a basis for the null space (kernel) of F .
10. (10 points) Suppose that the matrix
"
A=
5/2
#
1/2
1/2 −1/2
F
represents a linear operator R2 ←− R2 relative to the usual basis
(" # " #)
1
0
B=
,
0
1
for R2 . Find the matrix B that represents A with respect to the basis
(" # "
#)
1
1
S=
,
.
1
−1
11. Let
"
A=
7
4
4 13
#
.
(a) (5 points) Determine the eigen-values and associated eigen-vectors.
(b) (5 points) Find matrices P and D such that D is diagonal, P is invertible and
P −1 AP = D.
(c) (5 points) Compute A5 .
3
12. Consider the quadratic form
q(x, y) = 2x2 − 4xy + 8y 2 .
(a) (5 points) Give a (2 × 2)-symmetric matrix A such that
"
# " #
a b
x
q(x, y) = [x, y] ·
·
.
b d
y
(b) (5 points) Determine the eigen-values and associated eigen-vectors of A.
(c) (5 points) Find a diagonal matrix D and and invertible matrix P so that P −1 AP =
D.
(d) (5 points) Describe the quadratic curve q(x, y) = 1.
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