Math 423 Notes
3 November 2010
Section 4.5
Let’s look at the special product: A
T
A
1.
What is the rank (A
T
A)? rank(A), page 212
2.
What is the R(A
T
A)? R(A
T
) ***not doing this proof in class, may be used on a quiz or
test***
3.
What is the N(A
T
A)?
4.4.6 tells us that if the dimensions are the same then, N(A
T
A) = N(A)
Use: N(B)
⊂
N(AB), basically what we are trying to show is if you picture N(AB), N(B)
would be a subset or equal to N(AB).
Want to show that
x
ϵ
N(B)
x
ϵ
N(AB)
If
x
ϵ
N(B) –what information can I take from the definition of N(B)?
....
B
x
=
0
(definition of
nullspace)
B
x
=
0
AB
x
= A
0
AB
x
=
0
, therefore, N(B)
⊂
N(AB)
Now we can say: N(A)
⊂
N(A
T
A)
Looking at pg 198, 4.4.6, we have the assumption required to use 4.4.6.
If the dimensions of these two spaces are equal, dim N(A) = dim N(A
T
A), then we can make
the conclusion that the vector spaces are equal, N(A) = N(A
T
A)
Looking at pg 199:
dim N(A) = n  r
dim N(A) = n – rank (A)
dim N(A) = dim N(A
T
A) pg 199, 4.4.8
and
N(A) = N(A
T
A)
Given this as a fact: R(A
T
A) = R (A
T
)
Consider: A
x
=
b
, may/may not be consistent
A
T
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 Fall '10
 a
 Math, Linear Algebra, Vector Space, Dot Product, TA, best fit

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