Find all solutions of $Ax = 0$ in parametric vector form

Question

How can I find all solutions of $Ax = 0$ in parametric vector form where A is row equivalent to the matrix

$\begin{pmatrix} -1&-4&0&-4\\2&-8&0&8 \end{pmatrix}$

Answer 1

What you want to do first is put your matrix A into RREF (reduced row echelon form). The reduced form of your matrix A is: $\begin{pmatrix} 1&0&0&4\\0&1&0&0 \end{pmatrix}$ $\\$ So what can we do from here? We can write the general equation, which provides us with a parametric description of the solution set: $\\$

$x_{1}$ = -4$x_{4}$ $\\$

$x_{2}$ = 0 $\\$

$x_{3}$,$x_{4}$ are free $\\$

Now we can write our solution in parametric vector form: $\\$

$ \begin{pmatrix} x_{1}\\x_{2}\\x_{3}\\x_{4} \end{pmatrix}$ $ = \begin{pmatrix} -4x_{4}\\0\\x_{3}\\x_{4} \end{pmatrix}$ $ = x_{3} \begin{pmatrix} 0\\0\\1\\0 \end{pmatrix}$ $ + x_{4} \begin{pmatrix} -4\\0\\0\\1 \end{pmatrix}$

Answer 2

I think what your asking is to find the null space.

So follow these steps-

put your matrix to reduced row echleon form and then find the basis.

The span of the basis is the null space (all the solutions to Ax= 0)

Since you want to give it in a parametric vector form, it is the same thing as any linear combinations of the basis vector that you find.