Quotient space (linear algebra)

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 ${\displaystyle x\sim y}$ if ${\displaystyle x-y\in N}$. That is, ${\displaystyle x}$ is related to ${\displaystyle y}$ if one can be obtained from the other by adding an element of ${\displaystyle N}$. From this definition, one can deduce 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 often denoted

${\displaystyle [x]=x+N}$

since it is given by

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

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 well-defined (i.e. do not depend on the choice of representatives). 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]

There is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. The kernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence

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

If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may be constructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, the dimension of V is the sum of the dimensions of U and V/U. If V is finite-dimensional, it follows that the codimension of U in V is the difference between the dimensions of V and U:[6][7]

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

Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x in V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem for vector spaces says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank–nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank 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

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

When X is complete, then the quotient space X/M is complete with respect to the norm, and therefore a Banach space.

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

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• Dieudonné, Jean (1976), Treatise on Analysis, vol. 2, Academic Press, ISBN 978-0122155024
• Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces. Undergraduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-90093-4.
• Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9.
• Roman, Steven (2005). Advanced Linear Algebra. Graduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-24766-1.