MAS 3105
April 22, 2003
Final Exam
Prof. S. Hudson
Name
Show all your work and reasoning for maximum credit.
If you continue your work
on another page, be sure to leave a note. Do not use a calculator, book, or any personal
paper. You may ask about any ambiguous questions or for extra paper. If you use extra
paper, hand it in with your exam.
1) Prove that the product of two orthogonal matrices is also an orthogonal matrix.
2) Find an orthogonal or unitary diagonalizing matrix for
A
=
2
1
1
2
3) Suppose that
U
and
V
are subspaces of
W
. Show that
U
∩
V
is also a subspace.
4) Let
L
:
R
2
→
R
2
be a linear transformation. Assume
L
((2
,
0)
T
) = (2
,
6)
T
and
L
((0
,
3)
T
) = (6
,
12)
T
Find the matrix representation of
L
for the standard basis of
R
2
.
5) Answer True or False:
Every inconsistent linear system is overdetermined.
If det (A) is nonzero, then A is a product of elementary matrices.
Any two similar matrices must have the same eigenvalues.
If
A
is a 4x7 matrix then
Ax
= 0 has nontrivial solutions.
If
L
:
R
2
→
R
2
is linear, and
x
⊥
y
in
R
2
, then
L
(
x
)
⊥
L
(
y
).
Every square matrix is similar to some upper triangular matrix
T
.
If
A
is Hermitian then its diagonal entries (the
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Matrices, Diagonal matrix, Normal matrix, Det

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