4. Let A ∈ F^m×n
. Show that if y ∈ F^1×n
is in the row space of A and
x ∈ F^n×1
is in the null space of A, then yx = 0

Question

4. Let A ∈ F^m×n
. Show that if y ∈ F^1×n
is in the row space of A and
x ∈ F^n×1
is in the null space of A, then yx = 0

kinggeorge · Answer

Well, if $x\in \ker(A)$  ($\ker(A)$ is the null space), then $Ax=0$. Let $A=(\vec{a_1},\vec{a_2},...,\vec{a_n})$ where each $\vec{a_n}$ is the $n$-th row vector. Since $y$ is in the row space of A, we can say that $$y=r_1\vec{a_1}+...+r_n\vec{a_n}$$for some $r_1,...,r_n\in F$.Then. $$yx=r1\vec{a_1}x+...+r_n\vec{a_n}x.$$Since $Ax=0$, $\vec{a_j}x=0$ for all $1\le j\le n$, we can conclude that $yx=0$.

Did this make sense?

anonymous · Answer

Thanks so much