Question

Can anyone explain what the rank and nullity of a matrix are?

Answer 1

The rank is the number of linearly independent column vectors found in a matrix. For example:
\[\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]\]
Has a rank of 2 but:
\[\left[\begin{matrix}1 & 2 \\ 0 & 0\end{matrix}\right]\]
Because:
\[\left(\begin{matrix}2 \\ 0\end{matrix}\right)=2\left(\begin{matrix}1 \\ 0\end{matrix}\right)\]
So they are linearly dependent.
The nullity just means that the rank+nullity=number of columns. So in the first one the nullity would be zero because the rank is 2 and there are 2 columns and for the second one the nullity would be 1 because the rank is 1 and there are 2 columns (2-1=1)