Page 138 - Linear Algebra Done Right

Exercises
Find a polynomial q ∈P 2 (R) such that
24.
1
p( ) = 1 p(x)q(x) dx 125
2 0
for every p ∈P 2 (R).
25. Find a polynomial q ∈P 2 (R) such that
1 1
p(x)(cos πx) dx = p(x)q(x) dx
0 0
for every p ∈P 2 (R).
26. Fix a vector v ∈ V and define T ∈L(V, F) by Tu = u, v . For
a ∈ F, find a formula for T a.
∗
n
27. Suppose n is a positive integer. Define T ∈L(F ) by
T(z 1 ,...,z n ) = (0,z 1 ,...,z n−1 ).

Find a formula for T (z 1 ,...,z n ).
∗
28. Suppose T ∈L(V) and λ ∈ F. Prove that λ is an eigenvalue of T
¯
if and only if λ is an eigenvalue of T .
∗
29. Suppose T ∈L(V) and U is a subspace of V. Prove that U is
invariant under T if and only if U ⊥ is invariant under T .
∗
30. Suppose T ∈L(V, W). Prove that
(a) T is injective if and only if T ∗ is surjective;
(b) T is surjective if and only if T ∗ is injective.

31. Prove that

dim null T ∗ = dim null T + dim W − dim V

and
dim range T ∗ = dim range T

for every T ∈L(V, W).
32. Suppose A is an m-by-n matrix of real numbers. Prove that the
m
dimension of the span of the columns of A (in R ) equals the
n
dimension of the span of the rows of A (in R ).