## mathlife Group Title 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 one year ago one year ago

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?