MATH 603 Homework Set I
Due Monday, 22 September 2014
1. Let (V, h·, ·i) denote the vector space of all functions f (x) defined on the interval [−1, 1]
for which
Z 1
dx
f (x) √
<∞
1 − x2
−1
with inner product defined by
Z 1
dx
f (x)g(x) √
hf, gi =
.
1 − x2
−1
Apply the Gram-Schmidt Orthogonalization Procedure to the set B = {1, x, x2 } to
∗
construct an orthonormal set of vectors B = {e1 (x), e2 (x), e3 (x)}.
2. Let L : R2 → R2 denote the linear transformation whose matrix with respect to the
Natural Basis is
0 1
.
−1 0
Let B = {f1 , f2 } be the basis obtained from the natural basis by rotation through the
angle π/4. Find the matrix of L with respect to the basis B, [L]B .
3. Let L denote the transformation from R3
natural basis is

0

[L] = −2
2
→ R3 whose matrix with respect to the

2 −2
0
2 .
−2 0
Find Ran(L) and Nul(L).
4. Let B = {f1 , . . . , fn } be a basis for the inner product space (V, h·, ·i). Show that
[a ⊗ b]B = [a]B ⊗ [b]B F where F denotes the n × n matrix F = [hfi , fj i]. Also show
that [A∗ ]B = F −1 [A]TB F for any transformation A from V → V where A∗ denotes its
adjoint and [A]TB denotes the transpose of the matrix [A]B .
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