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1)showing that W is a subspace was easy (closed under addition and scalar multiplication)
2)I took [1 2 0] and [0 0 1] as a basis( dimension 2) because [ a 2a b] is a linear combination of these two but is it correct or should I've done something else?
3)for the third question I setup the system
1 0 1 0|0
1 0 0 1|0
0 1 0 0|0
and deduced the linearly independent vectors which form the basis for R3
so was my work correct?
for 2) you showed spanning, but not linear independence.
you only need three equations.
you want to find a 3rd vector (x,y,z) such that {(1,2,0),(0,0,1),(x,y,z)} is linearly independent.
one way to do this is to pick (x,y,z) so that it is orthogonal to (1,2,0) and (0,0,1).
if (1,2,0).(x,y,z) = 0, then x+2y = 0. that is: y = -x/2.
if (0,0,1).(x,y,z) = 0, then z = 0. can you combine these two conditions? prove the resulting set of 3 vectors is linearly independent.
the system you set up makes no sense at all, your vectors live in R^{3}, why do you have a matrix with 4 columns?