Editing Product (mathematics) (section)

==Products in linear algebra==
There are many different kinds of products in linear algebra. Some of these have confusingly similar names ([[outer product]], [[exterior product]]) with very different meanings, while others have very different names (outer product, tensor product, Kronecker product) and yet convey essentially the same idea. A brief overview of these is given in the following sections.

===Scalar multiplication===
{{main|Scalar multiplication}}
{{further|Scaling (geometry)}}

By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map <math>\R \times V \rightarrow V</math>.

===Scalar product===
{{main|Scalar product}}
A [[scalar product]] is a bi-linear map:

:<math>\cdot : V \times V \rightarrow \R </math>

with the following conditions, that <math>v \cdot v > 0</math> for all <math>0 \not= v \in V</math>.

From the scalar product, one can define a [[Norm (mathematics)|norm]] by letting <math>\|v\| := \sqrt{v \cdot v} </math>.

The scalar product also allows one to define an angle between two vectors:

:<math>\cos\angle(v, w) = \frac{v \cdot w}{\|v\| \cdot \|w\|}</math>

In <math>n</math>-dimensional Euclidean space, the standard scalar product (called the [[dot product]]) is given by:

:<math>\left(\sum_{i=1}^n \alpha_i e_i\right) \cdot \left(\sum_{i=1}^n \beta_i e_i\right) = \sum_{i=1}^n \alpha_i\,\beta_i</math>

===Cross product in 3-dimensional space===
{{main|Cross product}}
The [[cross product]] of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors.

The cross product can also be expressed as the [[formal calculation|formal]]{{Efn|Here, "formal" means that this notation has the form of a determinant, but does not strictly adhere to the definition; it is a mnemonic used to remember the expansion of the cross product.}} [[determinant]]:
:<math>\mathbf{u \times v} = \begin{vmatrix}
  \mathbf{i} & \mathbf{j} & \mathbf{k} \\
         u_1 &        u_2 &        u_3 \\
         v_1 &        v_2 &        v_3 \\
\end{vmatrix}</math>

===Composition of linear mappings===
{{main|Function composition}}
A linear mapping can be defined as a function ''f'' between two vector spaces ''V'' and ''W'' with underlying field '''F''', satisfying<ref>{{cite book|last1=Clarke|first1=Francis|title=Functional analysis, calculus of variations and optimal control|date=2013|publisher=Springer|location=Dordrecht|isbn=978-1447148203|pages=9–10}}</ref>
:<math>f(t_1 x_1 + t_2 x_2) = t_1 f(x_1) + t_2 f(x_2), \forall x_1, x_2 \in V, \forall t_1, t_2 \in \mathbb{F}.</math>

If one only considers finite dimensional vector spaces, then
:<math>f(\mathbf{v}) = f\left(v_i \mathbf{b_V}^i\right) = v_i f\left(\mathbf{b_V}^i\right) = {f^i}_j v_i \mathbf{b_W}^j,</math>
in which '''b<sub>V</sub>''' and '''b<sub>W</sub>''' denote the [[Basis (linear algebra)|bases]] of ''V'' and ''W'', and ''v<sub>i</sub>'' denotes the [[Tensor#Definition|component]] of '''v''' on '''b<sub>V</sub>'''<sup>''i''</sup>, and [[Einstein notation|Einstein summation convention]] is applied.

Now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping ''f'' map ''V'' to ''W'', and let the linear mapping ''g'' map ''W'' to ''U''. Then one can get
:<math>g \circ f(\mathbf{v}) = g\left({f^i}_j v_i \mathbf{b_W}^j\right) = {g^j}_k {f^i}_j v_i \mathbf{b_U}^k.</math>

Or in matrix form:
:<math>g \circ f(\mathbf{v}) = \mathbf{G} \mathbf{F} \mathbf{v},</math>
in which the ''i''-row, ''j''-column element of '''F''', denoted by ''F<sub>ij</sub>'', is ''f<sup>j</sup><sub>i</sub>'', and ''G<sub>ij</sub>=g<sup>j</sup><sub>i</sub>''.

The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.

===Product of two matrices===
{{main|Matrix product}}

Given two matrices

:<math>A = (a_{i,j})_{i=1\ldots s;j=1\ldots r} \in \R^{s\times r}</math> and <math>B = (b_{j,k})_{j=1\ldots r;k=1\ldots t}\in \R^{r\times t}</math>

their product is given by

:<math>B \cdot A = \left( \sum_{j=1}^r a_{i,j} \cdot b_{j,k} \right)_{i=1\ldots s;k=1\ldots t} \;\in\R^{s\times t}</math>

===Composition of linear functions as matrix product===
There is a relationship between the composition of linear functions and the product of two matrices. To see this, let r = dim(U), s = dim(V) and t = dim(W) be the (finite) [[dimension (mathematics)|dimensions]] of vector spaces U, V and W. Let 
<math>\mathcal U = \{u_1, \ldots, u_r\}</math> be a [[basis (linear algebra)|basis]] of U, 
<math>\mathcal V = \{v_1, \ldots, v_s\}</math> be a basis of V and 
<math>\mathcal W = \{w_1, \ldots, w_t\}</math> be a basis of W. In terms of this basis, let
<math>A = M^{\mathcal U}_{\mathcal V}(f) \in \R^{s\times r}</math>
be the matrix representing f : U → V and 
<math>B = M^{\mathcal V}_{\mathcal W}(g) \in \R^{r\times t}</math> 
be the matrix representing g : V → W. Then

:<math>B\cdot A = M^{\mathcal U}_{\mathcal W} (g \circ f) \in \R^{s\times t}</math>

is the matrix representing <math>g \circ f : U \rightarrow W</math>.

In other words: the matrix product is the description in coordinates of the composition of linear functions.

===Tensor product of vector spaces===
{{main|Tensor product}}

Given two finite dimensional vector spaces ''V'' and ''W'', the tensor product of them can be defined as a (2,0)-tensor satisfying:
:<math>V \otimes W(v, m) = V(v) W(w), \forall v \in V^*, \forall w \in W^*,</math>
where ''V<sup>*</sup>'' and ''W<sup>*</sup>'' denote the [[dual space]]s of ''V'' and ''W''.<ref>{{cite book|last1=Boothby|first1=William M.|title=An introduction to differentiable manifolds and Riemannian geometry|url=https://archive.org/details/introductiontodi0000boot|url-access=registration|date=1986|publisher=Academic Press|location=Orlando|isbn=0080874398|page=[https://archive.org/details/introductiontodi0000boot/page/200 200]|edition=2nd}}</ref>

For infinite-dimensional vector spaces, one also has the:
* [[Tensor product of Hilbert spaces]]
* [[Topological tensor product]].

The tensor product, [[outer product]] and [[Kronecker product]] all convey the same general idea.  The differences between these are that the Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its [[tensor (intrinsic definition)|intrinsic definition]]. The outer product is simply the Kronecker product, limited to vectors (instead of matrices).

===The class of all objects with a tensor product===
In general, whenever one has two mathematical [[object (category theory)|objects]] that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as the [[internal product]] of a [[monoidal category]].  That is, the monoidal category captures precisely the meaning of a tensor product; it captures exactly the notion of why it is that tensor products behave the way they do.  More precisely, a monoidal category is the [[class (set theory)|class]] of all things (of a given [[type theory|type]]) that have a tensor product.

===Other products in linear algebra===
Other kinds of products in linear algebra include:

* [[Hadamard product (matrices)|Hadamard product]]
* [[Kronecker product]]
* The product of [[tensor]]s:
** [[Exterior algebra|Wedge product or exterior product]]
** [[Interior product]]
** [[Outer product]]
** [[Tensor product]]