[Math] Suppose that $v_1,v_2,v_3,v_4$ spans $V$. Prove that the list $v_1 – v_2, v_2 – v_3, v_3-v_4,v_4$ also spans $V$.

linear algebra

Suppose that $v_1,v_2,v_3,v_4$ spans $V$. Prove that the list $v_1 – v_2, v_2 – v_3, v_3-v_4,v_4$ also spans $V$.

attempt: Suppose $v_1,v_2,v_3,v_4$ spans $V$,

then let $v \in V$.

So $v = a_1v_1 + a_2v_2 + a_3v_3 + a_4v_4$ for some $a_1,a_2,a_3,a_4 \in F$.

Then we will show $v \in span(v_1 – v_2, v_2 – v_3, v_3-v_4,v_4) $.

So $v = c_1(v_1 -v_2) + c_2(v_2-v_3) + c_3(v_3 – v_4) + c_4v_4$ for some $c_1,..,c_4 \in F.$.

Then setting the two equations equal we have $a_1v_1 + a_2v_2 + a_3v_3 + a_4v_4 = c_1(v_1 -v_2) + c_2(v_2-v_3) + c_3(v_3 – v_4) + c_4v_4$.

I am stuck and don't really know how to conclude. Can someone please help me?
Thank you

Best Answer

$\begin{eqnarray*}a_1v_1 + a_2v_2 + a_3v_3 + a_4v_4 &=& c_1(v_1 -v_2) + c_2(v_2-v_3) + c_3(v_3 - v_4) + c_4v_4\\&=&c_1v_1+(c_2-c_1)v_2+(c_3-c_2)v_3+c_4v_4\end{eqnarray*}$

Take $a_1=c_1, a_4=c_4, c_2=a_2+a_1$ and $c_3=a_3+a_2+a_1$. Then $$c_1(v_1 -v_2) + c_2(v_2-v_3) + c_3(v_3 - v_4) + c_4v_4=v.$$

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