Let V and W be the subspace of $RR^2$ spanned by (1,1) and (1,2),respectively. Find vectors $v ∈ V and w ∈ W$ so $v + w = (2,−1)$ ?

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Answer 1 · 2018-03-12T10:13:57

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If $vecv in V$ then $vecv=lambda(1,1)=(lambda,lambda)$

If $vecw in W$ then $vecw=rho(1,2)=(rho,2rho)$

$lambda, rho in RR$

Then $vecv+vecw=(lambda+rho,lambda+2rho)=(2,-1)$ Thus we have

$lambda+rho=2$
$lambda+2rho=-1$

The only solution is $lambda=5$ and $rho=-3$

Our vectors are $vecv=(5,5)$ and $vecw=(-3,-6)$

Let V and W be the subspace of RR^2RR^2 spanned by (1,1) and (1,2),respectively. Find vectors v ∈ V and w ∈ Wv ∈ V and w ∈ W so v + w = (2,−1)v + w = (2,−1)?