Editing Vector space (section)

==Basic constructions==
In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. 

===Subspaces and quotient spaces===
{{Main|Linear subspace|Quotient vector space}}
[[File:Linear subspaces with shading.svg|thumb|250px|right|A line passing through the [[origin (mathematics)|origin]] (blue, thick) in {{math|[[Euclidean space|'''R'''<sup>3</sup>]]}} is a linear subspace. It is the intersection of two [[plane (mathematics)|planes]] (green and yellow).]]

A nonempty [[subset]] <math>W</math> of a vector space <math>V</math> that is closed under addition and scalar multiplication (and therefore contains the <math>\mathbf{0}</math>-vector of <math>V</math>) is called a ''linear subspace'' of <math> V </math>, or simply a ''subspace'' of <math> V </math>, when the ambient space is unambiguously a vector space.{{sfn|Roman|2005|loc=ch. 1, p. 29}}<ref group=nb>This is typically the case when a vector space is also considered as an [[affine space]]. In this case, a linear subspace contains the [[zero vector]], while an affine subspace does not necessarily contain it.</ref> Subspaces of <math>V</math> are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set <math>S</math> of vectors is called its [[linear span|span]], and it is the smallest subspace of <math>V</math> containing the set <math>S</math>. Expressed in terms of elements, the span is the subspace consisting of all the [[linear combination]]s of elements of <math>S</math>.{{sfn|Roman|2005|loc=ch. 1, p. 35}}

{{anchor|vector line|vector plane|vector hyperplane}}Linear subspace of dimension 1 and 2 are referred to as a ''line'' (also ''vector line''), and a ''plane'' respectively. If ''W'' is an ''n''-dimensional vector space, any subspace of dimension 1 less, i.e., of dimension <math>n-1</math> is called a ''[[hyperplane]]''.{{sfn|Nicholson|2018|loc=ch. 10.4}}

The counterpart to subspaces are ''quotient vector spaces''.{{sfn|Roman|2005|loc=ch. 3, p. 64}} Given any subspace <math>W \subseteq V</math>, the quotient space <math>V / W</math> ("<math>V</math> [[modular arithmetic|modulo]] <math>W</math>") is defined as follows: as a set, it consists of
<math display="block">\mathbf{v} + W = \{\mathbf{v} + \mathbf{w} : \mathbf{w} \in W\},</math>
where <math>\mathbf{v}</math> is an arbitrary vector in <math>V</math>. The sum of two such elements <math>\mathbf{v}_1 + W</math> and <math>\mathbf{v}_2 + W</math> is <math>\left(\mathbf{v}_1 + \mathbf{v}_2\right) + W</math>, and scalar multiplication is given by <math>a \cdot (\mathbf{v} + W) = (a \cdot \mathbf{v}) + W</math>. The key point in this definition is that <math>\mathbf{v}_1 + W = \mathbf{v}_2 + W</math> [[if and only if]] the difference of <math>\mathbf{v}_1</math> and <math>\mathbf{v}_2</math> lies in <math>W</math>.<ref group=nb>Some authors, such as {{harvtxt|Roman|2005}}, choose to start with this [[equivalence relation]] and derive the concrete shape of <math>V / W</math> from this.</ref> This way, the quotient space "forgets" information that is contained in the subspace <math>W</math>.

The [[kernel (algebra)|kernel]] <math>\ker(f)</math> of a linear map <math>f : V \to W</math> consists of vectors <math>\mathbf{v}</math> that are mapped to <math>\mathbf{0}</math> in <math>W</math>.{{sfn|Lang|1987|loc=ch. IV.3.}} The kernel and the [[image (mathematics)|image]] <math>\operatorname{im}(f) = \{f(\mathbf{v}) : \mathbf{v} \in V\}</math> are subspaces of <math>V</math> and <math>W</math>, respectively.{{sfn|Roman|2005|loc=ch. 2, p. 48}} 

An important example is the kernel of a linear map <math>\mathbf{x} \mapsto A \mathbf{x}</math> for some fixed matrix <math>A</math>. The kernel of this map is the subspace of vectors <math>\mathbf{x}</math> such that <math>A \mathbf{x} = \mathbf{0}</math>, which is precisely the set of solutions to the system of homogeneous linear equations belonging to <math>A</math>. This concept also extends to linear differential equations
<math display=block>a_0 f + a_1 \frac{d f}{d x} + a_2 \frac{d^2 f}{d x^2} + \cdots + a_n \frac{d^n f}{d x^n} = 0,</math>
where the coefficients <math>a_i</math> are functions in <math>x,</math> too.
In the corresponding map
<math display=block>f \mapsto D(f) = \sum_{i=0}^n a_i \frac{d^i f}{d x^i},</math>
the [[derivative]]s of the function <math>f</math> appear linearly (as opposed to <math>f^{\prime\prime}(x)^2</math>, for example). Since differentiation is a linear procedure (that is, <math>(f + g)^\prime = f^\prime + g^\prime</math> and <math>(c \cdot f)^\prime = c \cdot f^\prime</math> for a constant <math>c</math>) this assignment is linear, called a [[linear differential operator]]. In particular, the solutions to the differential equation <math>D(f) = 0</math> form a vector space (over {{math|'''R'''}} or {{math|'''C'''}}).{{sfn|Nicholson|2018|loc=ch. 7.4}}

The existence of kernels and images is part of the statement that the [[category of vector spaces]] (over a fixed field <math>F</math>) is an [[abelian category]], that is, a corpus of mathematical objects and structure-preserving maps between them (a [[category (mathematics)|category]]) that behaves much like the [[category of abelian groups]].{{sfn|Mac Lane|1998}} Because of this, many statements such as the [[first isomorphism theorem]] (also called [[rank–nullity theorem]] in matrix-related terms)
<math display=block>V / \ker(f) \; \equiv \; \operatorname{im}(f)</math>
and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for [[group (mathematics)|groups]].

===Direct product and direct sum===
{{Main|Direct product|Direct sum of modules}}
The ''direct product'' of vector spaces and the ''direct sum'' of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.

The ''direct product''<!--explain direct--> <math>\textstyle{\prod_{i \in I} V_i}</math> of a family of vector spaces <math>V_i</math> consists of the set of all tuples <math>\left(\mathbf{v}_i\right)_{i \in I}</math>, which specify for each index <math>i</math> in some [[index set]] <math>I</math> an element <math>\mathbf{v}_i</math> of <math>V_i</math>.{{sfn|Roman|2005|loc=ch. 1, pp. 31–32}} Addition and scalar multiplication is performed componentwise. A variant of this construction is the ''direct sum'' <math display="inline">\bigoplus_{i \in I} V_i</math> (also called [[coproduct]] and denoted <math display="inline">\coprod_{i \in I}V_i</math>), where only tuples with finitely many nonzero vectors are allowed. If the index set <math>I</math> is finite, the two constructions agree, but in general they are different.

===Tensor product===
{{Main|Tensor product of vector spaces}}
The ''tensor product'' <math>V \otimes_F W,</math> or simply <math>V \otimes W,</math> of two vector spaces <math>V</math> and <math>W</math> is one of the central notions of [[multilinear algebra]] which deals with extending notions such as linear maps to several variables. A map <math>g : V \times W \to X</math> from the [[Cartesian product]] <math>V \times W</math> is called [[bilinear map|bilinear]] if <math>g</math> is linear in both variables <math>\mathbf{v}</math> and <math>\mathbf{w}.</math>  That is to say, for fixed <math>\mathbf{w}</math> the map <math>\mathbf{v} \mapsto g(\mathbf{v}, \mathbf{w})</math> is linear in the sense above and likewise for fixed <math>\mathbf{v}.</math>

[[Image:Universal tensor prod.svg|class=skin-invert-image|right|thumb|200px|[[Commutative diagram]] depicting the universal property of the tensor product]]
The tensor product is a particular vector space that is a ''universal'' recipient of bilinear maps <math>g,</math> as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called [[tensor]]s
<math display=block>\mathbf{v}_1 \otimes \mathbf{w}_1 + \mathbf{v}_2 \otimes \mathbf{w}_2 + \cdots + \mathbf{v}_n \otimes \mathbf{w}_n,</math>
subject to the rules{{sfn|Lang|2002|loc = ch. XVI.1}}
<math display=block>\begin{alignat}{6}
a \cdot (\mathbf{v} \otimes \mathbf{w}) ~&=~ (a \cdot \mathbf{v}) \otimes \mathbf{w} ~=~ \mathbf{v} \otimes (a \cdot \mathbf{w}), && ~~\text{ where } a \text{ is a scalar} \\
(\mathbf{v}_1 + \mathbf{v}_2) \otimes \mathbf{w} ~&=~ \mathbf{v}_1 \otimes \mathbf{w} + \mathbf{v}_2 \otimes \mathbf{w}  &&  \\
\mathbf{v} \otimes (\mathbf{w}_1 + \mathbf{w}_2) ~&=~ \mathbf{v} \otimes \mathbf{w}_1 + \mathbf{v} \otimes \mathbf{w}_2.  &&  \\
\end{alignat}</math>
These rules ensure that the map <math>f</math> from the <math>V \times W</math> to <math>V \otimes W</math> that maps a [[tuple]] <math>(\mathbf{v}, \mathbf{w})</math> to <math>\mathbf{v} \otimes \mathbf{w}</math> is bilinear. The universality states that given ''any'' vector space <math>X</math> and ''any'' bilinear map <math>g : V \times W \to X,</math> there exists a unique map <math>u,</math> shown in the diagram with a dotted arrow, whose [[function composition|composition]] with <math>f</math> equals <math>g:</math> <math>u(\mathbf{v} \otimes \mathbf{w}) = g(\mathbf{v}, \mathbf{w}).</math><ref>{{harvtxt|Roman|2005}}, Th. 14.3. See also [[Yoneda lemma]].</ref> This is called the [[universal property]] of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.