Page 395 - Matrix Analysis & Applied Linear Algebra

5.9 Complementary Subspaces 391

n×n
5.9.4. Explain why = S⊕ K, where S and K are the subspaces of
n × n symmetric and skew-symmetric matrices, respectively. What is

1 2 3
the projection of A = 4 5 6 onto S along K? Hint: Recall
7 8 9
Exercise 3.2.6.

5.9.5. Fora general vector space, let X and Y be two subspaces with respec-
tive bases B X = {x 1 , x 2 ,..., x m } and B Y = {y 1 , y 2 ,..., y n } .
(a) Prove that X∩ Y = 0 if and only if {x 1 ,..., x m , y 1 ,..., y n }
is a linearly independent set.
(b) Does B X ∪B Y being linear independent imply X∩ Y = 0?
(c) If B X ∪B Y is a linearly independent set, does it follow that X
and Y are complementary subspaces? Why?

5.9.6. Let P be a projector defined on a vector space V. Prove that (5.9.10)
is true—i.e., prove that the range of a projector is the set of its “fixed
points” in the sense that R (P)= {x ∈V | Px = x}.

5.9.7. Suppose that V = X⊕ Y, and let P be the projector onto X along
Y. Prove that (5.9.11) is true—i.e., prove

R (P)= N (I − P)= X and R (I − P)= N (P)= Y.

5.9.8. Explain why P ≥ 1 for every projector P = 0. When is P =1?
2 2
5.9.9. Explain why I − P = P for all projectors that are not zero and
2 2
not equal to the identity.

n×1 T
5.9.10. Prove that if u, v ∈ are vectors such that v u =1, then

I − uv T
= uv T
= u v = uv T
,

2 2 2 2 F
where is the Frobenius matrix norm defined in (5.2.1) on p. 279.
F
n
5.9.11. Suppose that X and Y are complementary subspaces of , and let
B =[X | Y]bea nonsingular matrix in which the columns of X and
Y constitute respective bases for X and Y. Foran arbitrary vector
n×1
v ∈ , explain why the projection of v onto X along Y can be
obtained by the following two-step process.
(1) Solve the system Bz = v for z.

z 1
(2) Partition z as z = , and set p = Xz 1 .
z 2