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Orthogonal subspaces
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Orthogonal subspaces
Solve
:
A
x
⃗
=
b
A\vec{x} = b
A
x
=
b

when there is no solution.
Now,
A
T
A
x
=
A
T
b
  
⟹
  
N
(
A
T
A
)
=
N
(
A
)
  
⟹
  
r
a
n
k
(
A
T
A
)
=
r
a
n
k
(
A
)
A^TAx = A^Tb \implies N(A^TA) = N(A) \implies rank(A^TA) = rank(A)
A
T
A
x
=
A
T
b
⟹
N
(
A
T
A
)
=
N
(
A
)
⟹
r
ank
(
A
T
A
)
=
r
ank
(
A
)

Thus, all rows of
A
A
A

must be linearly independent.
