Question

easy way to find the null space and column space of a matrix

Answer 1

The null space is the space of the corresponding homogeneous system of linear equations, so use Gauß' method. For the column space just take a maximal set of linearly independent column vectors from your matrix.

Answer 2

I unerstand the column space, but the null space is still getting me... I think its finding the free variables that i dont understand.

Answer 3

The null space is also called the kernel of the matrix and contains all those vectors which are mapped onto the 0 vector by the matrix. You can easily check, that it is a linear subspace. But where do you want to find free variables?

Answer 4

after row reduction we had to find the free variables...

Answer 5

Ah yes, so you are right inside of Gauß' elimination algorithm. After you finished row reduction, there may be some rows, that are zero. That means you can introduce free variables for those rows in the solution vector, which you can then use for backwards elimination.