Editing Linear algebra (section)

===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.-->