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 $M$.
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Furthermore, $V / M$ is considered to be endowed with the induced operations:
- $\paren {x + M} + {y + M} := \paren {x + y} + M$
- $\alpha \paren {x + M} := \alpha x + M$
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