span equality $span\{v_1,v_2\}=span\{av_1,bv_2\}$

Question

Prove for nonzero scalars a and b such that $$span\{v_1,v_2\}=span\{av_1,bv_2\}$$

My try:

$$span\{v_1,v_2\}=c_1v_1+c_2v_2=c'_1(av_1)+c'_2(bv_2)=span\{av_1,bv_2\}$$ by splitting contants. Is it correct to show ? then what is the role of non zero scalar word here ? i feel it is okay for 0 also

Answer 1

$span\{v_1,v_2\}=\{xv_1+yv_2:x,y\in\mathbb{K}\}=\{\frac{x}{a}(ax)+\frac{y}{b}(bv_2):x,y\in\mathbb{K}\}=\{x'(av_1)+y'(bv_2):x',y'\in\mathbb{K}\}=span\{av_1,bv_2\}$

Answer 2

Writing

$$span\{v_1,v_2\}=c_1v_1+c_2v_2$$ is not correct. It should be

$$span\{v_1,v_2\}=\{c_1v_1+c_2v_2 \mid (c_1,c_2) \in \mathbb F^2\}.$$

Then you could notice that for any $(c_1,c_2) \in \mathbb F^2$ you have

$$c_1v_1+c_2v_2= \frac{c_1}{a}(a v_1)+ \frac{c_2}{b}(b v_2)$$ as $a,b$ are not vanishing by hypothesis.