Simple sub-space question given solution vector $(1,2,3)$ , space($M_{3x3}(\Bbb R)$) and a system $Ax=0$
Solution 1:
Let$$S=\{A\in M_{3\times3}(\Bbb R)\mid A.(1,2,3)=0\}.$$Asserting that $M$ is a subspace of $M_{3\times3}(\Bbb R)$ means two thins: that the sum of the elements of $S$ belongs to $S$ and thay if $\lambda\in\Bbb R$ and if $A\in S$, then $\lambda A\in S$. But:
- If $A,B\in S$, then$$(A+B).(1,2,3)=A.(1,2,3)+B.(1,2,3)=0+0=0,$$and therefore $A+B\in S$.
- If $A\in S$ and $\lambda\in\Bbb R$, then$$(\lambda A)(1,2,3)=\lambda\bigl(A.(1,2,3)\bigr)=\lambda.0=0,$$and therefore $\lambda A\in S$.