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 $V/M$.

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra