Quotient Mapping is Linear Transformation

Theorem
Let $K$ be a field.

Let $V$ be a vector space over $K$.

Let $M$ be a subspace of $V$.

Let $V / M$ be the quotient vector space.

Let $Q: V \to V / M$ be the quotient mapping.

Then $Q$ is a linear transformation.

Proof
Let $\lambda, \mu \in K$ and $x, y \in V$.

Then we have:

using the definition of vector addition and scalar multiplication for $V/M$.