My understanding of the subspace still isn't solid enough, so I want to know what I know so far is at least correct.

By definition, a set of vectors S of R

^{n}is called a subspace of R

^{n}iff for all vectors (I will call them x):

1) (x+y) [tex]\in[/tex] S and

2) kx [tex]\in[/tex] S.

Also, the solution set of a homogeneous system is always a subspace.

When I encounter problems such as determine whether or not {(x

_{1},x

_{2},x

_{3})|x

_{1}x

_{2}=0} is a subspace of its corresponding R

^{n}, I would approach this problem as such:

Since the set has only three vectors, then it's in R

^{3}, first of all; then I check for 1): Suppose a set of vectors Y {(y

_{1},y

_{2},y

_{3})|y

_{1}y

_{2}=0}, then (S+Y)=x

_{1}x

_{2}+y

_{1}y

_{2}=0+0=0; for 2): kS=(kx

_{1})(kx

_{2})=(k0)(k0)=0. Therefore it is a subspace of R

^{3}.

Is my way of solving this problem correct?