STAT714 LINEAR STATISTICAL MODELS Fall Session 2009
Homework 3
Date given out: 9/14/09. Please submit your solutions to the problems to me by 12.20pm
9/21/09.
You may discuss homework problems with each other, but each student should write up his/her
solutions independently of others. Late homework will not be accepted (unless there is a very
good reason).
1. Suppose that Vis a vector space and that x1,x2, ..., xm∈ V.
(a) Prove that the set of all linear combinations of x1,x2, ..., xm; i.e.,
S=(x∈ V :x=
m
X
i=1
cixi)
is a subspace of V.
(b) Does {x1,x2, ..., xm}form a basis for S? If so, prove it. If not, provide a counterex-
ample.
2. Find all solutions to the system of equations
1111
1235
x1
x2
x3
x4
=0
1
3. Consider again the matrix from Homework 2, question 4:
A=
1 1 0 0
1 1 0 0
0 0 1 0
0 0 1 1
.
(a) Compute MA, the perpendicular projection matrix onto C(A).
(b) Compute I−MA, the perpendicular projection matrix onto N(AT).
4. (a) Suppose that Ais an n×nsymmetric matrix. Prove that Ais idempotent if and only
if r(A) + r(I−A) = n.
(b) Suppose that A,B, and A+Bare all idempotent. Prove that AB =0and BA =0.
5. Let Pbe an n×northogonal matrix and let Abe an n×nsymmetric and idempotent
matrix. Define D=PTAP.
Show that Dis a perpendicular projection matrix.