MAS 3105
March 15, 2007
Quiz 5 Key
Prof. S. Hudson
1) Find the matrix representation
A
of the linear operator
L
:
R
2
→
R
2
that reflects each
vector
x
about the line
x
2
=
x
1
and then projects it onto the
x
1
axis.
2) Answer TRUE or FALSE: Part a) refers to the ‘baby rabbit example’ from the lectures.
a) With 2 adults and 3 baby rabbits this year, we’ll have 9 rabbits total, next year.
b) The commuting diagram for Ch 4.3 (similarity) contains only one vector space,
V
.
c) If
A
is similar to itself, then
A
is nonsingular.
d) If
A
and
B
are row equivalent they have the same column space.
e) The matrix
A
that answers problem 1) has rank = 1.
3) Choose ONE of these.
a) Suppose
A
and
B
are nxn and
B
is nonsingular.
Prove that rank (
BA
) = rank
(
A
). [Hint: compare their nullspaces]
b) Suppose that
L
:
V
→
W
is linear and
S
is a subspace of
V
. Prove that
L
(
S
) is a
subspace of
W
.
c) State the formula in Thm 5.1.1 for
x
T
y
and prove it (in
R
2
).
Remarks and Answers:
The average was about 40/60.
The unofficial scale is:
A’s
4560, with 6 points per letter below that. I computed your semester average, based on
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Prof. S. Hudson, ‘baby rabbit example, previous sentence shows

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