Finding a basis of a vector set by finding redundant vectors

[Integrate]

There's a method in finding basis vectors that I don't understand conceptually.


Let me show an example.


#1

$$A = \begin{bmatrix} 1 & 2 & 3 & -1 \\ 3 & 5 & 8 & -2 \\ 1 & 1 & 2 & 0 \end{bmatrix}$$

$$B=\begin{bmatrix}1 & 0 & 1 & 1 \\0 & 1 & 1 & -1 \\0 & 0 & 0 & 0\end{bmatrix}$$
#2

$$\begin{bmatrix}x_1 \\x_2 \\x_3 \\x_4\end{bmatrix} = \begin{bmatrix}-x_3 - x_4 \\-x_3 + x_4 \\x_3 \\x_4\end{bmatrix}= x_3 \begin{bmatrix}-1 \\-1 \\1 \\0\end{bmatrix}+ x_4 \begin{bmatrix}-1 \\1 \\0 \\1\end{bmatrix}$$

#3 $$x_1 A_1 + x_2 A_2 + x_3 A_3 + x_4 A_4 = 0$$

$$(-2x_3 + x_4)A_1 + (3x_3 - 2x_4)A_2 + x_3A_3 + x_4A_4 = 0$$
#4 This is where I start getting lost, not that I was completely there to begin with.


We set $$x_{3}=1$$ $$x_{4}=0$$ and get
$$(-2)A_1 + 3A_2 + A_3 = 0$$

And we get $$2A_1 - 3A_2=A_3$$
#4

We then set $$x_{3}=0$$ $$x_{4}=1$$ and get $$-A_1 + 2A_2 = A_4$$


Because both these vectors can be created using A_1 and A_2 we remove them and deduce A_1 and A_2 is a basis set?



What are the concepts of this method, and why do we choose the values that we do?