Question

Show that if Ax is in the nullspace of A^T, then Ax=0

Answer 1

@ganeshie8 details about Ax and T ? please ---- ty.

Answer 2

For a vector \(v\) to be in a matrix \(B\)'s null space it must satisfy this equation by definition:

\[Bv=0\]

Similarly, \(Ax\) is a vector, since a matrix times a vector is a vector, and for it to be in the nullspace of \(A^\top\) that means:

\[A^\top (Ax) = 0\]

So far just definitions. The trick that took me a few minutes to figure out it, what's the length of the vector \(Ax\)?

Answer 3

\[Ker(A^T)=Im(A)^{orthogonal}\]

Answer 4

Awesome!
So that states kernel of A^T is orthogonal to column space

So if a vector exists in both column space and the kernel of A^T, then it must be the zero vector.
Ker(A^T) \(\cap\) Im(A) = zero vector