Math 237
Carter
Test 2 Fall 2010
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. Yield to pedestrians and bicyclists. Good luck.
1. Let V denote a vector space. Fill in the blanks: (5 points each)
(a) The span of the set {v1 , v2 , . . . , vk } ⊂ V is the set
{v ∈ V :
}
(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 } ⊂
V is said to be
.
of a matrix A ∈ M (m × n) is the span of the set of rows.
(c) The
(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) Let A be a given (3 × 2)-matrix. Explain why there is an equation AX = B
which has no solution.
3. (10 points) Show that the set of symmetric matrices S = {A ∈ M (n, n) : A = At } is a
subspace of M (n, n). What is its dimension? Describe a basis.
4. (10 points) Determine the image of the unit square (the square with vertices (0, 0)t ,
(1, 0)t , (1, 1)t , (0, 1)t ) under the linear transformation
" # "
#
x
3x − y
L
=
.
y
x − 2y
1
5. (10 points) Determine a basis for the null space of the matrix
"
#
3 2 0 2
.
0 2 3 −2
6. (25 points) The reduced row echelon form of

−1 −2

A=
2
 1
2
4
is

1 2 0
the matrix

8 1 3

4 1 −1 

2 1 −3
1
3
1
6

A0 = 
 0 0 1
0 0 0 0
− 53
1
6
0


.

(a) Give a basis for the column space of A.
(b) Give a basis for the row space of A.
(c) Give a basis for the null space of A.
(d) Write the vector [1, 1, 1]t as a linear combination of the basis vectors of the column
space that you found above.
(e) The column space defines a plane in M (3, 1). Give an equation for this plane in
the form Ax + By + Cz = 0 where A, B, and C are constants.
7. (10 points) A matrix A ∈ M (4, 7) has rank 2. What is the dimension of the null
space? Give an example of such a matrix, and a vector B ∈ M (4, 1) such that there
is no solution, X, to the equation AX = B. You may choose A to be in reduced row
echelon form.
2