Category:Quotient Vector Spaces

This category contains results about Quotient Vector Spaces.

Let $V$ be a vector space.

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

Then the quotient (vector) space of $V$ modulo $M$, denoted $V / M$, is defined as:

$\set {x + M : x \in X}$

where $x + M$ is the Minkowski sum of $x$ and $M$.

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Furthermore, $V / M$ is considered to be endowed with the induced operations:

$\paren {x + M} +_{V / M} \paren {y + M} := \paren {x + y} + M$

$\alpha \circ_{V / M} \paren {x + M} := \alpha x + M$

From Quotient Vector Space is Vector Space, it is shown that, $+_{V / M}$ and $\circ_{V / M}$ are well-defined, provided that $M$ is a vector subspace of $V$.

That is:

$\struct {V / M, +_{V / M}, \circ_{V / M} }$

is indeed a vector space, provided that $M$ is a vector subspace of $V$.