Union of two vector subspaces not a subspace?

Question

I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?

Answer 1

I think the issue here is that the union of two subspaces is not necessarily closed under addition, and subspaces need to be nonempty and closed under addition and scalar multiplication.

You can think of two nonparallel planes in R³ that each contain the origin. Each plane is a subspace of R³ and they intersect in a line through the origin. But when you add two nonzero vectors, one from each plane and neither of which lie on the intersecting line, the result will not lie in either plane, so you will not have closure.

Let P₁ = span{(1,0,0), (0,1,0)} = xy-plane, and let v₁ = (1,1,0) which is in P₁.

Let P₂ = span{(0,1,0), (0,0,1)} = yz-plane, and let v₂ = (0,1,1) which is in P₂.

Then v₁ + v₂ = (1,2,1) which is in neither the xy- nor yz-planes; hence the union of P₁ and P₂ is not closed under addition, and so is not a subspace.

A simpler example which is similar would be to consider two nonparallel lines through the origin as subspaces of R². Picking two nonzero points/vectors, one from each line, and then adding would yield a point/vector on neither line.

For yet another example, consider the vector space of polynomials up to degree three over R. Let A = {ax³ + bx where a,b are in R} and let B = {ax² where a is in R}. Then A and B are subspaces, but (x³ + x) + (x²) is not in A or B, and so is not in the union.