Question

I just need an example of a vector space that is closed under scalar multiplication but not under addition. That is all. Thanks for your wisdom.

Answer 1

Let {(x,y) in R²: y=2*x or y=3*x}. That's a pretty simple space: just two lines in R².

It's closed under scalar multiplication: if (x,y) lies on one of these lines (take y=3x, w.l.o.g.), then (αx,αy) is on the same line, since y=3ximpliesαy=3αx.

However, it's not closed under addition. If (x₁,y₁) satisfies y=2x and(x₂,y₂) satisfies y=3x, then(x₁+x₂,y₁+y₂) generally does not satisfy either of these relationships, so it is outside the space. Take(x₁=1,y₁=2) and(x₂=1,y₂=3) as anexample:(x₁+x₂=2,y₁+y₂=5), and5≠2*2 and 5≠3*2.