[Math] Why do row equivalent matrices have the same row space?

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so I understand row space of a matrix X to mean the subspace of $R^n$ spanned by the row vectors of A

Here is the text that is in question:

Can the rows of B be written as a linear combination of A just because they are row-reduced from A?

Say we have matrix:

$$\begin{bmatrix} 1 & 2 \\ 3 & 8 \end{bmatrix}$$
$$ -> \begin{bmatrix} 1 & 2 \\ 0 & 2 \end{bmatrix}$$

How is the row-reduced matrix a linear combination of A?

Answer 1

In your example, $(1,2)=(1,2)$ and $(0,2)=(3,8)-3(1,2)$. (You can see why this is always true from the definition of elementary row operations.)