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

Definition

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Formally, the construction is as follows. Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is related to y if one can be obtained from the other by adding an element of N. The quotient space V/N is then defined as V/~, the set of all equivalence classes over V by ~.

Let ^* denote the equivalence class containing x. We define scalar multiplication and addition on the equivalent classes by

• α= [αx for all α ∈ K, and
• y" target="_blank" >^*" target="_blank" >= [x+y.

It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). These operations turn the quotient space V/N into a vector space over K. They also turn the map $x\backslash mapsto*$ into an epimorphism.

Examples and properties

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This simplest example is to take a quotient of Rⁿ. Let m ≤ n and let R^m be the subspace spanned by the first m standard basis vectors. Two vectors of Rⁿ are then seen to be equivalent if and only if they are identical in the last n−m coordinates. The quotient space Rⁿ/ R^m is isomorphic to R^n−m in an obvious manner.

In general, if V is n-dimensional vector space and U is an m-dimensional subspace, the quotient space V/U has dimension n−m.

Let T : V → W be a linear operator. The kernel (or nullspace) of T, denoted ker(T) is the set of all x ∈ V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem of linear algebra 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 that 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).

Quotient of a Banach space by a subspace

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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

$\backslash |*\backslash |\_\{X/M\}\; =\; \backslash inf\_\{m\; \backslash in\; M\}\; \backslash |x-m\backslash |\_X.$

The quotient space X/M is complete with respect to the norm, so it is a Banach space.

Examples

Let Cdenote the Banach space of continuous real-valued functions on the interval ^*" target="_blank" >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.

See also

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• quotient set
• quotient group
• quotient module
• quotient space (in topology)

Linear algebra | Functional analysis

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