For the Fall 2010 Ma 237 class. I was looking through old tests, and I found two tests,
one from Spring 2009 and one from Fall 2000. Both are included below a list of suggested
problems for additional study. I will draw upon the current document as I write the next
test.
Observe in both cases (Sp09 and F00), the test was not computationally difficult. For the
upcoming test, I will try and preserve the computational ease, but if you have a calculator
that can compute the reduced row echelon form of a matrix (e.g. TI-83 or TI-89), then you
may use it.
Suggested Problems
• page 80-89 # 1, 14, 16, 22(c), 27, 30(a)
• page 107 # 1, 3
• page 120-127 # 1, 20,10-13 (also be able to address the situation in M (n, n) (see below)
• page 149-152 # 1, 5
• page 161-166 # 1, 6-9, 11-13,20
Symmetric and skew-symmetric matrices: In general, consider the sets S = {A ∈
M (n, n) : A = At }. and A = {A ∈ M (n, n) : A = −At }. Here, At denotes the transpose of A. Show that the dimension of S is 1 + 2 + 3 + · · · + n = n(n + 1)/2 while the
dimension of A is 1 + 2 + 3 + · · · + n − 1 = n(n − 1)/2. Observe that any A ∈ M (n, n) can
be written as the sum of a symmetric and a skew-symmetric matrix: let S = (A + At )/2,
and let B = (A − At )/2; then S t = (A + At )t /2 = (At + (At )t )/2=S; similarly; B = −B t ;
then A = S + B.
Required Definitions
1. The span of the set {v1 , v2 , . . . , vk } ⊂ V is the set
{v ∈ V : v =
k
X
αi vk for some αi ∈ R}
i=1
1
2. 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 linearly independent.
3. The row space of a matrix A ∈ M (m × n) is the span of the set of rows.
4. A basis for a vector space V is a set {v1 , v2 , . . . , vk } ⊂ V such that
(a) {v1 , v2 , . . . , vk } spans V in the sense that any vector v ∈ V , can be written as
P
v = ki=1 αi vk for some scalars αi ∈ R, and
(b) the set {v1 , v2 , . . . , vk } ⊂ Rn is linearly independent (see above).
5. If the vector space V has a basis {v1 , v2 , . . . , vk }, then the dimensions of V is k, that
is, the number of elements in a linearly independent set that spans V .
Math 237
Carter
Test 2 Spring 2009
General Instructions: Do not write on this test. Write your name on only the outside
of your blue books. Do all your work inside your blue books. Write neat complete answers
to the questions below. Show your work.
While popping pop-corn on the stove, you should heat the oil and throw in two or three
kernel. When the test kernels pop, you can add the rest. Cover the pan, but lift the top
slightly and occasionally to let the steam escape. Be careful not to burn the kernels, and try
not to leave too many uncooked seeds. Remove from heat immediately. Rely on your ears
and nose to tell you when it is done. Good luck.
1. Let A be a given (4×3)-matrix. Explain in complete sentences why there is an equation
AX = B which has no solution.
2. Determine the equation of the plane whose solution space is the span of the set of
vectors {[1, 1, 2]t , [1, 2, 0]t }.
3. Show that the set





a
b
c







U =  0 d e  : a, b, c, d, e, f ∈ R




 0 0 f

2
of upper triangular (3 × 3)-matrices is a subspace of the space M (3, 3). What is the
dimension of U?
4. Determine bases for the column space, row space, and null space of the matrix


1 2 −1 3



A =  2 2 −4 4 
.
1 3 0 4
5. Determine (draw) the image of the unit circle under the linear transformation
"
# that is
1 1
represented (with respect to the standard basis) by the matrix L =
.
0 1
"
#
a b
6. Let A =
denote a general (2 × 2)-matrix for which a 6= 0 and ad − bc 6= 0.
c d
Show, by using row reduction, the computation of the inverse matrix of A.
Math 237
Carter
Test 2 Fall 2000
General Instructions: Write your name on only the outside of your blue
book. Put your test paper inside your blue book as you leave. Solve each of
the following problems point values (out of 100 points) are indicated on the
problems.
1. Fill in the blanks: (5 points each)
(a) The span of the set {v1 , v2 , . . . , vk } ⊂ Rn is the set
{v ∈ Rn :
}
(b) 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 } ⊂ Rn is said to be
3
.
of a matrix A ∈ M (m × n) is the span of the set
(c) The
of rows.
(d) A basis for a vector space V is a set {v1 , v2 , . . . , vk } ⊂ V such that
i.
ii.
(e) If the vector space V has a basis {v1 , v2 , . . . , vk }, then the dimensions
of V is
.
2. (10 points each) For the following matrices compute the row space, column space, and null space.


1 0 4


2 0 7


(a) 

1 2 1


1 2 2
"
#
0 1 4 6 10
(b)
0 0 1 1 1


1 0 0 0


 0 −1 0 0 


(c) 

 0 0 0 −1 


0 0 1 0


1 1 −1 1 1



(d) 
 −1 1 −1 1 1 
0 0 0 0 1
3. (10 points each) Compute the inverses of the following matrices, A, and
solve the equation AX = B for the given vector B.
"
#
" #
1 −1
2
(a) A =
;B=
2 2
2
4


(b) A = 

1 1 2



5






;
B
=
2 1 1
 2
−2 2 1
−1
4. (15 points) Consider a matrix A ∈ M (5, 8). Suppose that the rank of A
is 4. What is the dimension of the null space? Is there a vector B ∈ R4
for which the equation AX = B has no solution? If no, then prove it. If
yes, then give an example of such a matrix A and a vector B
5