Let $A$ be an arbitrary $5 \times 5$ real matrix row equivalent to the following matrix $:$ $$R = \begin{pmatrix} 1 & 0 & 0 & -3 & -1 \\ 0 & 1 & 0 & -2 & -1 \\ 0 & 0 & 1 & -1 & -1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}$$
Then which of the following statement/s is/are true?
$(a)$ The first two columns of $A$ are linearly independent.
$(b)$ The last four columns of $A$ generate a space of dimension $3.$
I think when we reduce the rows of $A$ by elementary row operations to form $R$ then the independence of the columns remains unaltered and $(a)$ and $(b)$ are true for $A$ iff similar statements are true for $R.$ Am I right with my reasoning?