# Definition:Quotient Vector Space

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## Definition

Let $V$ be a vector space.

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

Then the **quotient 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 $M$.

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

- $\paren {x + M} + {y + M} := \paren {x + y} + M$
- $\alpha \paren {x + M} := \alpha x + M$

This article is complete as far as it goes, but it could do with expansion.In particular: So much more rich stuff to surround this but Conway is just glossing over preliminaries, not rigorously developing. Easiest way is probably to leverage the existing material on quotient structures to create versions for multiple operations in general, and vector spaces in particularYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also see

- Results about
**quotient vector spaces**can be found**here**.

## Sources

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