Question

3. Let A be an m X n = 0 and Aw = 0 matrix, and let v and w be vectors in R" with the property that Av Explain why A(v w) 0. Then explain why A(ev dw) 0 for each pair of scalars c and d.

Answer 1

Answer:

Let $A$ be an $m\times n$ matrix and $w, v$ vectors in $\mathbb{R}^n$ with the property that $Aw=0,\;Av=0$.

Then, using the distributive property between matrices we have that

$A(v+w)=Av+Aw=0+0=0$, so $A(v+w)=0$

Now, let c and d scalars. Observe that using the property of  product of a matrix by a scalar and the distributive property we have that

$A(cv+dw)=A(cv)+A(dw)=cAv+dAw=c*0+d*0=0$