Row equivalence. What is it exactly? When matrices are row equivalent... why is this important? If a matrix like:
$$\begin{bmatrix} 1 & 0 \\ -3 & 1 \end{bmatrix}$$
is row equivalent to the identity matrix (add 3 times the first row to the second), what does that mean exactly? Why is this a concept that we have to know as students of linear algebra? These matrices aren't equal.... they are row-equivalent. Why is this a useful concept to know?
 A: $$\begin{bmatrix} 1 & 0 &|&0\\ -3 & 1 &|&0\end{bmatrix}_{R_2\rightarrow R_2+3R_1}\tag{1}$$
$$\begin{bmatrix} 1 & 0 &|&0\\ 0 & 1 &|&0\end{bmatrix}_{  \mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }}\tag{2}$$
The above corresponding system of homogeneous equations convey the same information.
$$\begin{matrix}x=0&x=0\\-3x+y=0&y=0\end{matrix}$$
$\implies$ Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space.
$\implies$ Two matrices in reduced row echelon form have the same row space if and only if they are equal.
A: My reason for why the concept of row equivalence is important is that the solutions to the two matrix equations $$Ax=b$$ and $$Bx=b$$ are the same as long as $A$ and $B$ are row equivalent. Often time, you want to reduce an original metric equation $Ax=b$ to an equation $Bx=b$ that is easier to solve, where $B$ is row equivalent to $A$ since row operations do not change the solution set.
