Question

find a basis for all space of 2 by 3 matrices whose column add to zero?

Answer 1

Hint: If you know the first row of a matrix in this space, that determines the second row.

Answer 2

thank you einstein ;)

Answer 3

I think you mean that the sum of the elements of each column is zero. If that is what you want, the basis is:

\[\left[\begin{matrix}1 & 0 & 0\\ -1 & 0 & 0\end{matrix}\right]\]

\[\left[\begin{matrix}0 & 1 & 0\\ 0 & -1 & 0\end{matrix}\right]\]

\[\left[\begin{matrix}0 & 0 & 1\\ 0 & 0 & -1\end{matrix}\right]\]