Linear Algebra Solution set equal to Span

Question

I am confused on how to approach a certain question. It asks to find a collection of vectors $x_1,\ldots,x_p$ so that the solution set of equation $ax=0$ is equal to $\operatorname{span} \{x_1,\ldots x_p\}$

They give a matrix of what $A$ is. What I am doing at the moment is creating a augmented matrix with $A$ and zeros in the last column. After I row reduce the matrix, I get $x_2=-x_4$ and $x_3=0$. How can I get the solution set from my answer?

Answer 1

Could you specify, what $A$ is or write the row-reduced form of $A$? Then your question will be clearer. Given that you have got the row-reduced form, you are just a step away from getting the vectors $x_1,\ldots, x_p$ such that they span the solution space of $AX = 0$

Answer 2

You're seeking a basis for the null space of $A$. If I understand correctly, you've reached the reduced row echelon form augmented matrix: $$ \left[\begin{array}{cccc|c} 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \end{array}\right] $$ (possibly with some extra rows of zeroes). The solutions to this system of linear equations (i.e., the null space) has the form $$\mathrm{null}(A)=\{(\alpha,\beta,0,-\beta):\alpha,\beta \in K\}$$ (where $K$ is the underlying field).

We could alternatively write this $$\mathrm{null}(A)=\{\alpha(1,0,0,0)+\beta(0,1,0,-1):\alpha,\beta \in K\}$$ which should give you an idea of what's a good choice of basis for the null space.