True or false (and why) The subset S of R^2 consisting of all vectors of the form(a,1)form a subspae of R^2?

False.


For S to be a subspace it needs to satisfy a few requirements that it doesn't. Three examples:


- We must have that if x and y are in S, then x+y is in S. But clearly (1,1) and (0,1) are in S, but their sum (1,2) is not in S.


- Similarly, we need that for every t in R and x in S, t*x is in S. But (1,1) is in S and 2*(1,1)=(2,2) is not.


- S does not contain the zero vector, which is another requirement for a subspace.