Editing Linear algebra (section)

==Duality==
{{main|Dual space}}
A [[linear form]] is a linear map from a vector space {{mvar|V}} over a field {{mvar|F}} to the field of scalars {{mvar|F}}, viewed as a vector space over itself. Equipped by [[pointwise]] addition and multiplication by a scalar, the linear forms form a vector space, called the '''dual space''' of {{mvar|V}}, and usually denoted {{mvar|V*}}<ref>{{Harvp|Katznelson|Katznelson|2008}} p. 37 §2.1.3</ref> or {{mvar|V{{prime}}}}.<ref>{{Harvp|Halmos|1974}} p. 20, §13</ref><ref>{{Harvp|Axler|2015}} p. 101, §3.94</ref>

If {{math|'''v'''<sub>1</sub>, ..., '''v'''<sub>''n''</sub>}} is a basis of {{mvar|V}} (this implies that {{mvar|V}} is finite-dimensional), then one can define, for {{math|1=''i'' = 1, ..., ''n''}}, a linear map {{math|''v<sub>i</sub>''*}} such that {{math|''v<sub>i</sub>''*('''v'''<sub>''i''</sub>) {{=}} 1}} and {{math|''v<sub>i</sub>''*('''v'''<sub>''j''</sub>) {{=}} 0}} if {{math|''j'' ≠ ''i''}}. These linear maps form a basis of {{math|''V''*}}, called the [[dual basis]] of {{math|'''v'''<sub>1</sub>, ..., '''v'''<sub>''n''</sub>}}. (If {{mvar|V}} is not finite-dimensional, the {{math|''v<sub>i</sub>''*}} may be defined similarly; they are linearly independent, but do not form a basis.)

For {{math|'''v'''}} in {{mvar|V}}, the map
:<math>f\to f(\mathbf v)</math>
is a linear form on {{mvar|V*}}. This defines the [[canonical map|canonical linear map]] from {{mvar|V}} into {{math|(''V''*)*}}, the dual of {{mvar|V*}}, called the '''[[double dual]]''' or '''[[bidual]]''' of {{mvar|V}}. This canonical map is an [[isomorphism]] if {{mvar|V}} is finite-dimensional, and this allows identifying {{mvar|V}} with its bidual. (In the infinite-dimensional case, the canonical map is injective, but not surjective.)

There is thus a complete symmetry between a finite-dimensional vector space and its dual. This motivates the frequent use, in this context, of the [[bra–ket notation]]
:<math>\langle f, \mathbf x\rangle</math>
for denoting {{math|''f''('''x''')}}.

===Dual map===
{{main|Transpose of a linear map}}

Let 
:<math>f:V\to W</math>
be a linear map. For every linear form {{mvar|h}} on {{mvar|W}}, the [[composite function]] {{math|''h'' ∘ ''f''}} is a linear form on {{mvar|V}}. This defines a linear map
:<math>f^*:W^*\to V^*</math>
between the dual spaces, which is called the '''dual''' or the '''transpose''' of {{mvar|f}}.

If {{mvar|V}} and {{mvar|W}} are finite-dimensional, and {{mvar|M}} is the matrix of {{mvar|f}} in terms of some ordered bases, then the matrix of {{mvar|f*}} over the dual bases is the [[transpose]] {{math|''M''<sup>T</sup>}} of {{mvar|M}}, obtained by exchanging rows and columns.

If elements of vector spaces and their duals are represented by column vectors, this duality may be expressed in [[bra–ket notation]] by 
:<math>\langle h^\mathsf T , M \mathbf v\rangle = \langle h^\mathsf T M, \mathbf v\rangle.</math>
To highlight this symmetry, the two members of this equality are sometimes written 
:<math>\langle h^\mathsf T \mid M \mid \mathbf v\rangle.</math>

===Inner-product spaces===
{{Main|Inner product space}}

Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as an [[inner product]]. The inner product is an example of a [[bilinear form]], and it gives the vector space a geometric structure by allowing for the definition of length and angles. Formally, an ''inner product'' is a map.

:<math> \langle \cdot, \cdot \rangle : V \times V \to F </math>

that satisfies the following three [[axiom]]s for all vectors {{math|'''u''', '''v''', '''w'''}} in {{math|''V''}} and all scalars {{math|''a''}} in {{math|''F''}}:<ref name= Jain>{{Cite book|title=Functional analysis|author=P. K. Jain, Khalil Ahmad|chapter-url=https://books.google.com/books?id=yZ68h97pnAkC&pg=PA203|page=203|chapter=5.1 Definitions and basic properties of inner product spaces and Hilbert spaces|isbn=81-224-0801-X|year=1995|edition=2nd|publisher=New Age International}}</ref><ref name="Prugovec̆ki">{{Cite book|title=Quantum mechanics in Hilbert space|author=Eduard Prugovec̆ki|chapter-url=https://books.google.com/books?id=GxmQxn2PF3IC&pg=PA18|chapter=Definition 2.1|pages=18 ''ff''|isbn=0-12-566060-X|year=1981|publisher=Academic Press|edition=2nd}}</ref>
* [[complex conjugate|Conjugate]] symmetry:
*:<math>\langle \mathbf u, \mathbf v\rangle =\overline{\langle \mathbf v, \mathbf u\rangle}.</math>
:In <math>\mathbb{R}</math>, it is symmetric.
* [[Linear]]ity in the first argument:
*:<math>\begin{align}
\langle a \mathbf u, \mathbf v\rangle &= a \langle \mathbf u, \mathbf v\rangle. \\
\langle \mathbf u + \mathbf v, \mathbf w\rangle &= \langle \mathbf u, \mathbf w\rangle+ \langle \mathbf v, \mathbf w\rangle.
\end{align}</math>
* [[Definite bilinear form|Positive-definiteness]]:
*:<math>\langle \mathbf v, \mathbf v\rangle \geq 0</math>
:with equality only for {{math|'''v''' {{=}} 0}}.

We can define the length of a vector '''v''' in ''V'' by
:<math>\|\mathbf v\|^2=\langle \mathbf v, \mathbf v\rangle,</math>
and we can prove the [[Cauchy–Schwarz inequality]]:
:<math>|\langle \mathbf u, \mathbf v\rangle| \leq \|\mathbf u\| \cdot \|\mathbf v\|.</math>

In particular, the quantity
:<math>\frac{|\langle \mathbf u, \mathbf v\rangle|}{\|\mathbf u\| \cdot \|\mathbf v\|} \leq 1,</math>
and so we can call this quantity the cosine of the angle between the two vectors.

Two vectors are orthogonal if {{math|⟨'''u''', '''v'''⟩ {{=}} 0}}. An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. Given any finite-dimensional vector space, an orthonormal basis could be found by the [[Gram–Schmidt]] procedure. Orthonormal bases are particularly easy to deal with, since if {{nowrap|1='''v''' = ''a''<sub>1</sub> '''v'''<sub>1</sub> + ⋯ + ''a<sub>n</sub>'' '''v'''<sub>''n''</sub>}}, then
:<math>a_i = \langle \mathbf v, \mathbf v_i \rangle.</math>

The inner product facilitates the construction of many useful concepts. For instance, given a transform {{math|''T''}}, we can define its [[Hermitian conjugate]] {{math|''T*''}} as the linear transform satisfying
:<math> \langle T \mathbf u, \mathbf v \rangle = \langle \mathbf u, T^* \mathbf v\rangle.</math>
If {{math|''T''}} satisfies {{math|''TT*'' {{=}} ''T*T''}}, we call {{math|''T''}} [[Normal matrix|normal]]. It turns out that normal matrices are precisely the matrices that have an orthonormal system of eigenvectors that span {{math|''V''}}.<!-- This is a potentially useful remark, but a proper context needs to be set for it. One can say quite simply that the [[linear]] problems of [[mathematics]]—those that exhibit [[linearity]] in their behavior—are those most likely to be solved. For example, [[differential calculus]] does a great deal with linear approximation to functions. The difference from [[nonlinearity|nonlinear]] problems is very important in practice.-->