w = {( x , y , 5x -6y) : x and y are real numbers } is subspace of V= R³

Question

 How we can prove that is a subspace ?

Answer 1

If A is a subspace of B (A⊆B) then from  ∀x∈A follows that x∈B as well. But not vice sersa: ∃y∈B, such that y∉A.

So you have to take an arbitary element from W and show that it belongs to V and find at least one element which belongs to V but not W. I think that is easy to do.

Since all elements of W are 3 dimensional vectors, then any element from W belongs to V as well. But elements of W cannot be any vector from V due to the third coordinate of w,  5x -6y; the third is a "connection" (between) of the first and the second coordinates. Trry to find one vector and you are done. I hope this will help.