Assignment 6
1) Let W be a subspace of the inner product space V and let :fVW→be the
function which maps a vector to its orthogonal projection onto W. So, if
where , then
1
uuu=+
212
and uW u W∈⊥
( )
1
fuu=
.
a. Prove that
f
is a linear transformation.
b. Prove that
f
ff=. (This property of functions is called idempotence).
2) Suppose that U is an orthogonal (orthonormal) matrix. Prove that
and conclude that
()()
Ux Uy x y=
ii Ux x=
and Ux Uy⊥
if and only if
x
y
⊥
. (Recall that to prove an “if and only if” theorem, you must prove two
separate directions.)
3) Fit the best line to the given data and calculate the least squares error.
a) (0,l), (1,1), (2,2), (3,2)
b) (-1,0), (0,1), (1,2), (2,4)
c) (1,7), (-1,0), (-2,7)
4) Find the orthogonal matrix which diagonalizes and determine .
P A 1
PAP
−
a)
311
131
113
A
⎡⎤
⎢⎥
=⎢⎥
⎢⎥
⎣⎦
b)
724 0 0
24 7 0 0
00 724
00247
A
−
⎡⎤
⎢⎥
⎢⎥
=⎢⎥
−
⎢⎥
⎣⎦
c)
211
12 1
112
A
−−
⎡⎤
⎢⎥
=− −
⎢⎥
⎢⎥
−−
⎣⎦
d) , 0
ab
Ab
ba
⎡⎤
=≠
⎢⎥
⎣⎦
5) Suppose that A is an matrix. Prove that has an orthonormal set of
eigenvectors.
mn×T
AA
6) Suppose that where P is orthogonal and R is upper triangular. Prove
that if A is symmetric, then R is also symmetric and, hence, diagonal.
1
A PRP−
=
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