Null space of a matrix A..

Question

I was wondering, Should we have to change matrix A to a row reduced echelon matrix necessarily always when finding the Null space? Isnt just an echelon form not enough to calculate the Null space,its basis and dimensions?

Thanks for any help.

Answer 1

In fact, all 3 matrices $A$, the echelon form of $A$ and row reduced echelon form of $A$ have the same null space. But to characterize it exactly is most simple when the matrix is in row reduced echelon form.

Answer 2

It's not necessary to reduce a matrix $A$ to its row reduced echelon to find its null space. You can do it from echelon form. In certain cases you don't even need to reduce it at all. For example, if you have a square invertible matrix (if you can easily see the columns are linearly independent) then the null space is trivial. Similarly, other observations can be used to determine the null space of a linear transformation without explicitly finding its row reduction form.