Images and preimages of subspaces under linear maps

Proposition. Let $T:X\to Y$ be a linear operator between vector spaces.

1. If $M\subset X$ is a linear subspace , then $T(M)\subset Y$ is a linear subspace.
2. If $N\subset Y$ is a linear subspace, then the preimage $T^{-1}(N):=\{x\in X\mid T(x)\in N\}$ is a linear subspace of $X$.

In particular, $\ker T$ and $\operatorname{im}T$ from image/kernel are subspaces.

Proof sketch. For (1), use $T(m_1+m_2)=T(m_1)+T(m_2)$ and $T(\lambda m)=\lambda T(m)$ and closure of $M$. For (2), if $x_1,x_2\in T^{-1}(N)$ then $T(x_1+x_2)\in N$ by closure of $N$; similarly for scalars.