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MATH 245 Linear Algebra 2, Assignment 2
Due Wed Oct 6
1:
Let
U
and
V
be vector spaces in
R
n
.
(a) Show that (
U
+
V
)
⊥
=
U
⊥
∩
V
⊥
.
(b) Show that (
U
∩
V
)
⊥
=
U
⊥
+
V
⊥
.
(c) Given that
U
= Col(
A
) and
V
= Col(
B
), ﬁnd a matrix
C
such that (
U
+
V
)
⊥
= Null(
C
).
(d) Given that
U
= Null(
A
) and
V
= Null(
B
), ﬁnd a matrix
C
such that (
U
∩
V
)
⊥
= Col(
C
).
2:
For a vector space
U
in
R
n
, we deﬁne the
reﬂection
in
U
to be the map Reﬂ
U
:
R
n
→
R
n
given by
Reﬂ
U
(
x
) =
x

2 Proj
U
⊥
(
x
)
.
(a) Let
{
u
1
,u
2
,
···
,u
k
}
be a basis for
U
and let
A
=
(
u
1
,u
2
,
···
,u
k
)
∈
M
n
×
k
(
R
). Find the matrix of Reﬂ
U
in terms of
A
.
(b) Show that Reﬂ
U
preserves length, that is
±
±
Reﬂ
U
(
x
)
±
±
=

x

for all
x
∈
R
n
.
3:
For two aﬃne spaces
P
and
Q
in
R
n
, the
distance
between
P
and
Q
is deﬁned to be
dist(
P,Q
) = min
n
dist(
x,y
)
±
±
±
x
∈
P,y
∈
Q
o
.
(a) Let
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