Exam II
APPM 3310
Spring 2002
Show your work or explain your reasoning for each answer.
1. Suppose
1
0
1
0
,
1
0
−1
2
is a basis for a subspace Vof R4, and
let V⊥denote the orthogonal complement of V.
(a) What is the dimension of V⊥? (Show the calculation, explain the
reasoning.)
(b) Find a basis for V⊥.
(c) Construct the matrix of projection onto V⊥.
2. A matrix Pis a projection matrix if and only if it satisfies two proper-
ties.
(a) What are the two properties?
(b) If Pis a projection matrix, show that P1=I−Pis also a pro-
jection matrix. (Hint: Show that P1satisfies the two properties
you gave in answer to the previous question. If you do not know
these properties, I will sell them to you for 5 points each.)
3. Suppose {α1, α2}is a basis for a subspace Wof R4, where
α1=
1
0
2
1
, α2=
1
−1
0
1
.
Construct an orthonormal basis {q1, q2}for W.
4. For x, y ∈Rn, show that
1
2kx+yk2+kx−yk2=kxk2+kyk2.