Suppose that $X$ is a vector space and $A$ is a subset of $X$. Let $Y$ be a proper subspace of $X$. Consider the collection
$M=\{a+Y:~a\in A\}.$
It is given that the collection $M$ is linear. In fact,
$M=\{x+Y:~x\in X\}.$
From this can we conclude that $A$ is a subspace of $X$? If not then what structure $A$ should possess? A detailed answer (or Hint or example or counter example) would be of much help. Thanks in advance.
EDIT We may assume that $A$ is closed under scalar multiplication.