Quotient space (linear algebra)

In linear algebra, the quotient of a vector space ${\displaystyle V}$ by a subspace ${\displaystyle N}$ is a vector space obtained by "collapsing" ${\displaystyle N}$ to zero. The space obtained is called a quotient space and is denoted ${\displaystyle V/N}$ (read "${\displaystyle V}$ mod ${\displaystyle N}$" or "${\displaystyle V}$ by ${\displaystyle N}$").

Formally, the construction is as follows.^[1] Let ${\displaystyle V}$ be a vector space over a field ${\displaystyle \mathbb {K} }$, and let ${\displaystyle N}$ be a subspace of ${\displaystyle V}$. We define an equivalence relation ${\displaystyle \sim }$ on ${\displaystyle V}$ by stating that ${\displaystyle x\sim y}$ iff ${\displaystyle x-y\in N}$. That is, ${\displaystyle x}$ is related to ${\displaystyle y}$ if and only if one can be obtained from the other by adding an element of ${\displaystyle N}$. This definition implies that any element of ${\displaystyle N}$ is related to the zero vector; more precisely, all the vectors in ${\displaystyle N}$ get mapped into the equivalence class of the zero vector.

The equivalence class – or, in this case, the coset – of ${\displaystyle x}$ is defined as

${\displaystyle [x]:=\{x+n:n\in N\}}$

and is often denoted using the shorthand ${\displaystyle [x]=x+N}$.

The quotient space ${\displaystyle V/N}$ is then defined as ${\displaystyle V/_{\sim }}$, the set of all equivalence classes induced by ${\displaystyle \sim }$ on ${\displaystyle V}$. Scalar multiplication and addition are defined on the equivalence classes by^[2][3]

• ${\displaystyle \alpha [x]=[\alpha x]}$ for all ${\displaystyle \alpha \in \mathbb {K} }$, and
• ${\displaystyle [x]+[y]=[x+y]}$.

It is not hard to check that these operations are

). These operations turn the quotient space ${\displaystyle V/N}$ into a vector space over ${\displaystyle \mathbb {K} }$ with ${\displaystyle N}$ being the zero class, ${\displaystyle [0]}$.

The mapping that associates to ${\displaystyle v\in V}$ the equivalence class ${\displaystyle [v]}$ is known as the quotient map.

Alternatively phrased, the quotient space ${\displaystyle V/N}$ is the set of all affine subsets of ${\displaystyle V}$ which are parallel to ${\displaystyle N}$.^[4]

Let X = R² be the standard

which only intersects the line at the origin.)

Another example is the quotient of Rⁿ by the subspace spanned by the first m

to R^n−m in an obvious manner.

Let ${\displaystyle {\mathcal {P}}_{3}(\mathbb {R} )}$ be the vector space of all cubic polynomials over the real numbers. Then ${\displaystyle {\mathcal {P}}_{3}(\mathbb {R} )/\langle x^{2}\rangle }$ is a quotient space, where each element is the set corresponding to polynomials that differ by a quadratic term only. For example, one element of the quotient space is ${\displaystyle \{x^{3}+ax^{2}-2x+3:a\in \mathbb {R} \}}$, while another element of the quotient space is ${\displaystyle \{ax^{2}+2.7x:a\in \mathbb {R} \}}$.

More generally, if V is an (internal) direct sum of subspaces U and W,

${\displaystyle V=U\oplus W}$

then the quotient space V/U is naturally isomorphic to W.^[5]

An important example of a functional quotient space is an L^p space.

There is a natural

${\displaystyle 0\to U\to V\to V/U\to 0.\,}$

If U is a subspace of V, the

${\displaystyle \mathrm {codim} (U)=\dim(V/U)=\dim(V)-\dim(U).}$

Let T : V → W be a

of T).

The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).

If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on X/M by

${\displaystyle \|[x]\|_{X/M}=\inf _{m\in M}\|x-m\|_{X}=\inf _{m\in M}\|x+m\|_{X}=\inf _{y\in [x]}\|y\|_{X}.}$

Let C[0,1] denote the Banach space of

. Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1]/M is isomorphic to R.

If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.

The quotient of a

{p_α | α ∈ A} where A is an index set. Let M be a closed subspace, and define seminorms q_α on X/M by

${\displaystyle q_{\alpha }([x])=\inf _{v\in [x]}p_{\alpha }(v).}$

Then X/M is a locally convex space, and the topology on it is the

quotient topology

.

If, furthermore, X is

See also

• Quotient group
• Quotient module
• Quotient set
• Quotient space (topology)

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