Matrix Theory, Math6304
Lecture Notes from August 30, 2012
taken by Andy Chang
Last Time (8/28/12)
Course info:
website  math.uh.edu/
∼
bgb
Matrix multiplication:
left (premultiplication) and right (postmultiplication)
Nullity and rank:
”dimension counting”
Dot product and orthogonality:
inner product, orthogonal projection, least squares property
GramSchmidt:
 set of linearly independent vectors can yield an orthonormal set that spans
the space
1
Further Review
1.1
GramSchmidt (cont’d)
1.1.9 Proposition
(GramSchmidt)
.
Given a linearly independent set
{
x
1
, x
2
, . . . , x
n
}
in
C
m
,
then there exists an orthonormal system
{
z
1
, z
2
, . . . , z
n
}
such that for each
j
≤
n
,
span
{
x
l
:
l
≤
j
}
= span
{
z
l
:
l
≤
j
}
.
Proof.
Since orthonormality implies linear independence, the dimension of both sides is equal.
It is enough to show that we can find inductively for each
j
≤
n
a vector
z
j
which forms an
orthonormal system
{
z
1
, z
2
, . . . , z
j
}
with the preceding ones and
z
j
=
u
j,
1
x
1
+
u
j,
2
x
2
+
· · ·
+
u
j,j
x
j
,
with appropriate coe
ﬃ
cients
u
j,k
, so
z
j
∈
span
{
x
l
:
l
≤
j
}
. As a consequence,
z
k
∈
span
{
x
l
:
l
≤
j
}
for
k
≤
j
and thus
span
{
z
l
:
l
≤
j
}
⊂
span
{
x
l
:
l
≤
j
}
but since the dimension on both sides is equal, the two spans must be the same.
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 Fall '12
 BernhardBodmann
 Linear Algebra, Multiplication, Matrices, Counting, Det

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