Mathematics 2940
Practice Questions for Prelim 1
Fall 2010
These are typical questions, perhaps more than you would expect on a single prelim.
1. Let
A
=
2
1
3
0
1

1
2
1
4

1
7
2
and
b
=
5
0
4
.
(a) Find all solutions to the equation
A
x
=
b
.
(b) Find all solutions to the equation
A
x
=
0
.
(c) Find a basis for the nullspace and the column space of A, and state the dimension
of each of these spaces.
2. Let
A
=
1
0
1
0
1
1
0
0

2
.
(a) Show that
A
is invertible and find
A

1
.
(b) Express
A

1
as a product of elimination matrices.
(c) Use your answer to (a) to find a 3
×
3 matrix
X
such that
XA
=
1
0
1
0
1
0
1
0
1
.
(d) Determine the
LU
or
PLU
factorization, as appropriate, of the matrix
A
+
A
T
.
3.
(a) Let
W
be the set of vectors in
R
4
orthogonal (perpendicular) to all of the vectors
(0
,
1
,

2
,
3), (1
,
1
,

4
,
2), and (2
,

1
,
0
,

4). Show that
W
is a subspace of
R
4
.
(b) If
u
,
v
, and
w
are any vectors in
R
4
, show that there is a nonzero vector orthog
onal to all of them.
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 '05
 HUI
 Math, Linear Algebra, Algebra, equation Ax

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