Page 137 - Linear Algebra Done Right

Chapter 6. Inner-Product Spaces
124
Find an orthonormal basis of P 2 (R) (with inner product as in
14.
Exercise 10) such that the differentiation operator (the operator
that takes p to p )on P 2 (R) has an upper-triangular matrix with

respect to this basis.
15. Suppose U is a subspace of V. Prove that
dim U ⊥ = dim V − dim U.
16. Suppose U is a subspace of V. Prove that U ⊥ ={0} if and only if
U = V.
17. Prove that if P ∈L(V) is such that P 2 = P and every vector
in null P is orthogonal to every vector in range P, then P is an
orthogonal projection.

2
18. Prove that if P ∈L(V) is such that P = P and
Pv ≤ v
for every v ∈ V, then P is an orthogonal projection.
19. Suppose T ∈L(V) and U is a subspace of V. Prove that U is
invariant under T if and only if P U TP U = TP U .
20. Suppose T ∈L(V) and U is a subspace of V. Prove that U and
U ⊥ are both invariant under T if and only if P U T = TP U .
4
21. In R , let

U = span (1, 1, 0, 0), (1, 1, 1, 2) .
Find u ∈ U such that u − (1, 2, 3, 4) is as small as possible.
22. Find p ∈P 3 (R) such that p(0) = 0, p (0) = 0, and

1
2
|2 + 3x − p(x)| dx
0
is as small as possible.

23. Find p ∈P 5 (R) that makes
π
2
| sin x − p(x)| dx
−π
as small as possible. (The polynomial 6.40 is an excellent approx-
imation to the answer to this exercise, but here you are asked to
find the exact solution, which involves powers of π. A computer
that can perform symbolic integration will be useful.)