Page 135 - Linear Algebra Done Right

Chapter 6. Inner-Product Spaces
122
Exercises
2
1. Prove that if x, y are nonzero vectors in R , then
x, y = x y cos θ,
where θ is the angle between x and y (thinking of x and y as
arrows with initial point at the origin). Hint: draw the triangle
formed by x, y, and x − y; then use the law of cosines.
2. Suppose u, v ∈ V. Prove that u, v = 0 if and only if

u ≤ u + av
for all a ∈ F.

3. Prove that
n 2 n n 2

2 b j
a j b j ≤ ja j
j
j=1 j=1 j=1
for all real numbers a 1 ,...,a n and b 1 ,...,b n .
4. Suppose u, v ∈ V are such that
u = 3, u + v = 4, u − v = 6.
What number must v equal?
2
5. Prove or disprove: there is an inner product on R such that the
associated norm is given by

(x 1 ,x 2 ) =|x 1 |+|x 2 |

2
for all (x 1 ,x 2 ) ∈ R .
6. Prove that if V is a real inner-product space, then
2 2
u + v −Hu − v
u, v =
4
for all u, v ∈ V.

7. Prove that if V is a complex inner-product space, then
2
2
2
2
u + v −Hu − v + u + iv i −Hu − iv i
u, v =
4
for all u, v ∈ V.