Math 4404 Projection Operators
Hand in Thursday, July 20
1. Read sections 1.9-1.12 from Saad’s book.
2. Compute the matrix Pwhich solves the projection problem for M= span{v1, v2}
and S=M⊥where
v1=1
2
1
1
1
1
v2=1
2
1
−1
−1
1
.
Once you have the matrix P, demonstrate that Px = 0 for x∈Sby choosing xto
be a linear combination of the basis vectors of S, i.e. let x=αw1+βw2where w1
and w2span S.
3. Compute the kernel of
A=1 7
7 49 .
4. In R3, let M= span
1
0
0
,
0
1
0
and S=M⊥. Then for v=
7
2
8
,
(a) Compute what Sis.
(b) Compute the projection operator Pwhich projects onto Mand solves the pro-
jection problem.
(c) Project vonto Mvia the matrix P.
(d) Compute v−P v and show it is orthogonal to Mby taking the inner product
with a linear combination of M’s basis vectors.
(e) Plot M,v,P v, and v−P v. Notice how v−P v is actually perpendicular to M.
(f) Compute ker(P).
(g) Show that R3= ran(P)⊕ker(P). To do this, show that for any vector x∈R3
that we can show there are vectors v1∈ran(P) and v2∈ran(I−P) such that
x=v1+v2.
5. Is the projector computed in problem 1 above an orthogonal or oblique projector?
Prove your assertion.
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