Linear algebra: Prove or disprove

[Baron]

Prove or disprove: N(A^T) = N(U^T ) where U is the row reduced echelon form of matrix A

I think its true. Here's my proof

Proof:

A = LU where L is an invertible matrix as the series of row operations to row reduce A can be written as an invertible matrix

If y ϵ N(A^T) then A^T y = 0

A = LU --> A^T y = U^T L^T y = 0

as L is invertible the range of L^T y is everything including y

so U^T y = 0 and y ϵ N(U^T). Hence N(A^T) is a subset of N(U^T)

I know I have to prove the other direction but is this okay so far? I'm a bit unsure about the "L is invertible, range is everything part"

 

[HallsofIvy]

Yes, that is correct- although "everything" is a bit vague! It might be better to say that "the range of L is then entire space". Also it would have helped if you had stated at the beginning that "N(A)" was the nullity of A.