# 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:

 $\ds \map Q {\lambda x + \mu y}$ $=$ $\ds \paren {\lambda x + \mu y} + M$ $\ds$ $=$ $\ds \paren {\lambda x + M} + \paren {\mu y + M}$ $\ds$ $=$ $\ds \lambda \paren {x + M} + \mu \paren {y + M}$ $\ds$ $=$ $\ds \lambda \map Q x + \mu \map Q y$

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

$\blacksquare$