# Describe all vectors orthogonal to col(A) with a twist

Tags:
1. Nov 3, 2015

### Bill Thompson

I am trying to solve the following problem:

Let A be a real mxn matrix. Describe the set of all vectors in F^m orthogonal to Col(A).

Here, F^m could be C^m. Now in the real case, I'd say that the column space of A is the row space of A^T, and it is well known that the row space of a matrix is orthogonal to it's null space ---> Col(a) is orthogonal to Null(A^T) (left nullspace). After significant research, I can't see how to change/adapt this statement for the complex field. Any help is greatly appreciated.

Thanks

2. Nov 3, 2015

### RUber

I don't think much changes when you move to the complex numbers, or any field in general.
If some real vectors $\{R^m\}$ are orthogonal to the columns of a purely real A, then the imaginary vectors $\{i R^m\}$ will also be orthogonal since it is a scalar multiple of an orthogonal vector.
So, then you essentially should have any linear combination of those real and imaginary vectors.