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MATH 51 FINAL EXAM
(December 6, 2004)
1.
Consider the matrices
A
=
0 0 1 1
1 3 1 2
2 6 1 3
and
R
=
1 3 0 1
0 0 1 1
0 0 0 0
.
The matrix
R
is the row reduced echelon form of
A
. (You do not need to check
this.)
1(a).
Find a basis for the column space of
A
.
1(b).
Find a basis for the null space of
R
.
1(c).
Note that
A
1
1
1
1
=
2
7
12
.
Find all solutions to
A
x
=
2
7
12
.
2.
Consider the following system of equations:
x
2
+
x
3
=
a
x
1
+
x
2
+ 2
x
3
=
b
x
1
+ 2
x
2
+ 3
x
3
=
c
2
x
1
+ 3
x
2
+ 5
x
3
=
d
Find the condition(s) on
a
,
b
,
c
, and
d
, for the system to have a solution. (Your
answer should be one or more equations of the form ?
a
+?
b
+?
c
+?
d
=?.)
3(a).
Find all eigenvalues of the matrix
A
=
1 0 2
7 3 5
2 0 1
.
3(b).
Consider the matrix
M
=
3 1 1
0 4 1
0 0 3
. Note that
v
1
=
1
1
0
is an eigenvec-
tor with eigenvalue 4. Find eigenvectors
v
2
and
v
3
so that
v
1
,
v
2
, and
v
3
form a
basis for
R
3
.

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