Linear Algebra and Its Applications · Problem 3.1.11

Orthogonality — Gilbert Strang

Mathematics · orthogonal-vectors-subspaces

Problem

The fundamental theorem is often stated in the form of Fredholm's alternative: For any

A

and

b

, one and only one of the following systems has a solution: ((i))

Ax = b

. ((ii))

A^Ty = 0

y^Tb \neq 0

. Either

b

is in the column space

C(A)

or there is a

y

N(A^T)

such that

y^Tb \neq 0

. Show that it is contradictory for (i) and (ii) both to have solutions.