Assignment 5
1)
a) Let A be an invertible n n× matrix. For vectors and u v
in n
R, define
(
)
(
)
, :
A
u v Au Av
=
i
Prove that this gives an inner product on
n
R
.
b) Let
cos sin
:sin cos
A
θ θ
θ θ
−
=
. Prove that ,A
u v u v
=
i
c) (Bonus) Give a geometric explanation of why the inner product in b)
works this way.
2) Calculate the inner product of the given vectors using the given inner product:
a) On
3
R
using the dot product,
1 5
2 7
3 9
i and
1 2
0 4
1 2
−
i.
b) If
(
)
2
2 1 0
p x p x p x p
= + +
and
(
)
2
2 1 0
q x q x q x q
= + +
, define
0 0 1 1 2 2
, :
p q p q p q p q
= + +
Calculate 2 2 2
, and 3x 2 5,5 2 1
x x x x x
+ − + +
.
c) On
[
]
0, 2
C let 0
2
, : ( ) ( )
f g f x g x dx
=∫. Calculate 2
,
x x
e e
and
2 2
3 2 5,5 2 1
x x x x
+ − + +
.
d) (Bonus) Consider the standard basis
{
}
2
1, ,
x x
on the space of polynomials
in b). With respect to this basis, the inner product defined in b) is strongly
related to the dot product in
3
R
. How?
3) Prove the triangle inequality for any inner product space:
u v u v
+ ≤ +
(Hint: for any real numbers a and b,
a b a b
+ ≤ +
.)
4)
a) Let
1
1
1
:u
=
and let
0
1
1
:v
−
−
=
. Using the dot product on
3
R
, find the
orthogonal projection of
u
onto
v
.
b) Use a) to find the part of
u
which is perpendicular to
v
. (i.e.
1 2
u u u
= +
where 1
u kv
=
for some real number k, and 2
0
u v
=
i
)
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