I'm having a tough time finding any resources online or even in my book that give me a good way to solve problems like the one I have. I have a couple like it, so if someone could explain generally while explaining the given example, it would be greatly appreciated. The question is
"Determine a spanning set for the null space of the given matrix A."
A = {{1,2,3,5},{1,3,4,2},{2,4,6,-1}}.
Seems like it would be a simple enough problem, I know some of what to do but it doesn't seem systematic or reliable, and it hasn't worked for every problem, so I must be doing something wrong. Thanks in advance everyone.
Best Answer
First reduce it to RREC: $$ A' = \begin{bmatrix} 1 && 0 && 1 && 0 \\ 0 && 1 && 1 && 0 \\ 0 && 0 && 0 && 1 \end{bmatrix} $$ Now lets find vectors $v = [w, x, y, z]$ that satisfy $vA = 0$ $$ w + y = 0 $$ $$ x + y = 0 $$ $$ z = 0 $$ Then, solving for $w, x, z$: $w = -y, x = -y, z = 0$.
Hence $[-y, -y, y, 0] \in \mathrm{Null}(A)$. From that you can conclude $\mathrm{span}([-1, -1, 1, 0]) = \mathrm{Null}(A)$.