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

See below

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)#