Question 1034158
Let L be any line in {{{R^3}}} that passes through the origin.  Then it would have the symmetric equation {{{x/a = y/b = z/c}}} with <a,b,c> as its direction vector.
Let w and v be two vectors in L.  Then w = k(a,b,c) and v = l(a,b,c) for some constants k and l.
Now a non-empty subset of any vector space is a subspace iff {{{alpha*w+beta*v}}} is also in the subset for any two vectors w and v in the said subset.

But {{{alpha*w+beta*v }}}= {{{alpha}}}*k(a,b,c) + {{{beta}}}*l(a,b,c)

= ({{{alpha}}}*k + {{{beta}}}*l)(a,b,c),

meaning the resulting linear combination is still in the line L.

Hence any line through the origin of {{{R^3}}} is a subspace of {{{R^3}}}