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Math 20F Bender
Final Exam
3 PM June 15, 1995
Please start each problem on a NEW PAGE.
Remember to show your work.
1. (25 pts.) Deﬁne the following:
(a) an eigenvalue of the
n
×
n
matrix
A
,
(b) the null space of an
m
×
n
matrix
A
,
(c) the dimension of a vector space
V
.
2. (15 pts.)
Show that
T
(
x
1
,x
2
3
)=(3
x
2

x
3
1

4
x
2
3
) is a linear transformation and ﬁnd its
standard matrix.
3. (25 pts.) Find the eigenvalues and eigenspaces of
11 2
02

1
00

1
.
4. (20 pts.) Let
W
be the span of the orthogonal vectors
1
1
0

1
,
1
0
1
1
and
0

1
1

1
. Write
3
4
5
6
as the
sum of a vector in the subspace
W
and a vector orthogonal to
W
.
5. (20 pts.)
Find the eigenvalues associated with the quadratic form 8
x
2
1
+6
x
1
x
2
and use them to
classify the form.
6. (60 pts.) Suppose that
A
is an
n
×
n
matrix and that
R
is the reduced row echelon form of
A
.Y
ou
are given
R
but
are not given
A
. For each of the following explain why you can or cannot answer it
given
R
but not
A
.
(a) Does
A

1
exist?
(b) What is a basis for Nul
A
?
(c) What is a basis for Col
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This note was uploaded on 06/18/2009 for the course MATH 20E taught by Professor Enright during the Winter '07 term at UCSD.
 Winter '07
 Enright
 Math, Linear Algebra, Algebra

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