We are to do Gauss-Jordan Elimination, and find the appropriate coefficients. My textbook gives this example:
Let $\boldsymbol{u_1} = (1, 0, 1)$, $\boldsymbol{u_2} = (1, 1, 2)$ and $\boldsymbol{u_3} = (-1, 2, 1)$.
Also, let $\boldsymbol{v_1} = (1, 2, 3)$ and $\boldsymbol{v_2} = (2, -1, 1)$.
Now, to prove $\text{span}\{\boldsymbol{u_1}, \boldsymbol{u_2}, \boldsymbol{u_3}\} \subseteq \text{span}\{\boldsymbol{v_1}, \boldsymbol{v_2}\}$, we apply Gauss-Jordan Elimination to the following augmented matrix:
$$ \left( \begin{array}{cc|c|c|c} 1&2&1&1&-1\\ 2&-1&0&1&2\\ 3&1&1&2&1 \end{array} \right) \rightarrow \left( \begin{array}{cc|c|c|c} 1&0&\frac{1}{5}&\frac{3}{5}&\frac{3}{5}\\ 0&1&\frac{2}{5}&\frac{1}{5}&-\frac{4}{5}\\ 0&0&0&0&0 \end{array} \right)$$
And therefore $\boldsymbol{u_1} = \frac{1}{5}\boldsymbol{v_1} + \frac{2}{5}\boldsymbol{v_2}$, $\boldsymbol{u_2} = \frac{3}{5}\boldsymbol{v_1} + \frac{1}{5}\boldsymbol{v_2}$, and $\boldsymbol{u_3} = \frac{3}{5}\boldsymbol{v_1} - \frac{4}{5}\boldsymbol{v_2}$.
I understand that the coefficients of $\boldsymbol{v_1}$ and $\boldsymbol{v_2}$ correspond to the values in the third, fourth and fifth columns of the resultant reduced row-echelon form matrix; but I still don't understand why those values correspond to the coefficients of $\boldsymbol{v_1}$ and $\boldsymbol{v_2}$ for $\boldsymbol{u_1}$, $\boldsymbol{u_2}$ and $\boldsymbol{u_3}$.
Could I have an explanation for this?
When we solve the equation $$Ax=b$$
If we let $A_i$ be the $i$-th column of $A$.
this is equivalent to $$\sum_{i=1}^n A_i x_i = b$$
That is the solution $x$ tells us how to express $b$ as linear combination of $A_i$.