Page 136 - Linear Algebra Done Right

Exercises
A norm on a vector space U is a function : U → [0, ∞) such
8.
that u = 0 if and only if u = 0, αu =|α| u for all α ∈ F
and all u ∈ U, and u + v ≤ u + v for all u, v ∈ U. Prove 123
that a norm satisfying the parallelogram equality comes from
an inner product (in other words, show that if is a norm
on U satisfying the parallelogram equality, then there is an inner
1/2
product , on U such that u = u, u for all u ∈ U).
9. Suppose n is a positive integer. Prove that This orthonormal list is
often used for
1 sin x sin 2x sin nx cos x cos 2x cos nx

√ , √ , √ ,..., √ , √ , √ ,..., √ modeling periodic
2π π π π π π π
phenomena such as
is an orthonormal list of vectors in C[−π, π], the vector space of tides.
continuous real-valued functions on [−π, π] with inner product
π
f, g = f(x)g(x) dx.
−π

10. On P 2 (R), consider the inner product given by
1
p, q = p(x)q(x) dx.
0
2
Apply the Gram-Schmidt procedure to the basis (1,x,x ) to pro-
duce an orthonormal basis of P 2 (R).
11. What happens if the Gram-Schmidt procedure is applied to a list
of vectors that is not linearly independent?
12. Suppose V is a real inner-product space and (v 1 ,...,v m ) is a
linearly independent list of vectors in V. Prove that there exist
exactly 2 m orthonormal lists (e 1 ,...,e m ) of vectors in V such
that
span(v 1 ,...,v j ) = span(e 1 ,...,e j )
for all j ∈{1,...,m}.

13. Suppose (e 1 ,...,e m ) is an orthonormal list of vectors in V. Let
v ∈ V. Prove that

2 2 2
v =| v, e 1 | +· · ·+| v, e m |
if and only if v ∈ span(e 1 ,...,e m ).