Additive Cancellation law for vector subspace

Question

If $A,B$ and $C$ are vector subspaces, then prove that if $A+B=C+B$, then $A=C$ or give counterexample to invalidate this.

I am not sure it this is true or not. I know that for the equality to hold, $A+B\subset C+B$ and $C+B\subset A+B$ should be true. Can anyone help?

Answer 1

Let $A = C$ and $B = 0$ then:

$$C = C + C = A+ C = B + C = 0 + C = C$$

And $A\neq B$ as long as $C \neq 0$ (where $0$ is the trivial vector space).

Answer 2

Counterexample: $B = \mathrm {span } (e_1,e_2,e_3), A = \mathrm {span} (e_1), C = \mathrm {span} (e_3)$, where $e_1,e_2,e_3$ are linearly independent. Then $A+B =C +B$ by defintion.