Name:
CSU ID:
Homework 11
November 20, 2015
The problems below were from HW10. We had to catch up, so these problems were assigned again.
1. Let hx, yi be the standard inner product <3 . For ~x = [−3, 2, 1]T , ~y =
[7, 1, 5]T , and ~z = [−1, 0, −3]T . Find the following
(a) ||~x||.
(b) ||~x − ~y ||, ||~y − ~z||.
(c) the angle between ~x and ~z.
(d) ||~x − ~y ||.
(e) Angles between ~x, ~y , and ~z.
2. Consider the weighted inner product in <3 defined by h~x, ~y i = 2x1 y1 +
3x2 y2 + x3 y3 . For the vectors in the previous problem, repeat the
computations for this inner product.
3. In the previous problem, it was claimed that definition was an inner
product. Show this; i.e., for arbitrary ~u, ~v , w
~ and constant k show that
(a) h~u, ~v i = h~v , ~ui
(b) h~u + ~v , wi
~ = h~u, wi
~ + h~u, wi.
~
(c) hk~u, ~v i = kh~u, ~v i.
(d) h~u, ~ui ≥ 0 and h~u, ~ui = 0 if an only if ~u = ~0.
4. A general class of inner products on <n is called matrix inner products.
~
For a given nonsingular matrix A it can be shown that h~v , ~v i = A~u ·AV
defines an inner product. (a) Verify each condition when A is given
below. (b) For ~v = [2, −1]T , find ~u such that h~v , ~ui = 0. (The answer
is not uniques.)
"
#
−3 4
A=
2 1
5. Let ~u = [u1 , u2 ]T and ~v = [v1 , v2 ]T . Find a matrix that generates the
inner product on <2 .
(a) h~u, ~v i = 3u1 v1 + 5u2 v2
(b) h~u, ~v i = 4u1 v1 + 6u2 v2
6. Given the inner products in the previous problem, and ~u = [2, −3]T , ~v =
[5, 6]T , complete the following
(a) ||~u||, ||~v ||
(b) ||2~u − ~v ||
(c) Find the angle between ~u and ~v .
7. Show that the following identity hold for vectors in any inner product
space.
||~u + ~v ||2 + ||~u − ~v ||2 = 2||~u||2 + 2||~v ||2
8. The standard inner product on Pn , polynomials of degree less than or
equal to n for p~ = p(x) = a0 +a1 x+· · · an xn , ~q = q(x) = b0 +b1 x+n xn
is given below. Complete the following.
(a) Find the angle between p~ = p(x) = 1 + x + x2 − 2x3 and ~q =
q(x) = 4 − x + 3x2 .
(b) Find a polynomial, r(x) of degree less than or equal to 3 such
that it is orthogonal to both p, q.
hp, qi =
n
X
aj bj
j=0
9. Let the vector space P2 have the inner product
h~
p, ~qi =
Z 1
p(x)q(x)dx
−1
(a) Find ||~
p|| for p~ = x, and p~ = x2 .
(b) Find the distance between p~ = 1 and ~q = x.
(c) Find an ~r in P2 such that it is orthogonal to p~ = 1 and ~q = x.