Name:
CSU ID:
Homework 10
November 13, 2015
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.