Question 1034158: Show that a line through the origin of R^3 is a subspace of R^3
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Let L be any line in  that passes through the origin. Then it would have the symmetric equation  with 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  is also in the subset for any two vectors w and v in the said subset.
But  =  *k(a,b,c) +  *l(a,b,c)
= (*k + *l)(a,b,c),
meaning the resulting linear combination is still in the line L.
Hence any line through the origin of is a subspace of
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