Math 235  Sample Final 2
NOTE:
The questions on this exam does not exactly reflect which questions will be on this terms
exam. That is, some questions asked on this exam may not be asked on our exam and there may
be some questions on our exam not asked here.
1.
Short Answer Problems
a) Define the four fundamental subspaces of a matrix
A
.
b) Let
A
=
3
i

i
2
. Is
A
normal?
c) Show that the product of two orthogonal matrices is an orthogonal matrix.
d) State the Fundamental Theorem of Linear Algebra.
2.
Let
W
be the vector space of all 2
×
2 upper triangular matrices with real entries. So
W
=
a
b
0
c

a, b, c
∈
R
.
Consider the linear mapping
L
:
R
3
→
W
defined by
L
(
x, y, z
) =
x
+
y
x
+
z
0
y

z
.
a) Find the rank and nullity of
L
.
b) Explain why
L
is not an isomorphism. Explain why
R
3
and
W
are isomorphic
without finding an isomorphism between them.
c) Find the matrix of
L
with respect to the standard basis,
S
, for
R
3
and the basis for
W
B
=
1
0
0
0
,
0
1
0
0
,
0
0
0
1
3.
Let
V
be an inner product space, with inner product
h
,
i
. Let
B
=
{
~v
1
, . . . ,~v
n
}
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 Spring '08
 CELMIN
 Math

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