Editing Inner product space (section)

==Related products==

The term "inner product" is opposed to [[outer product]] ([[tensor product]]), which is a slightly more general opposite. Simply, in coordinates, the inner product is the product of a <math>1 \times n</math> {{em|covector}} with an <math>n \times 1</math> vector, yielding a <math>1 \times 1</math> matrix (a scalar), while the outer product is the product of an <math>m \times 1</math> vector with a <math>1 \times n</math> covector, yielding an <math>m \times n</math> matrix. The outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the {{em|[[Trace (linear algebra)|trace]]}} of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is vertical times horizontal and expands out".

More abstractly, the outer product is the bilinear map <math>W \times V^* \to \hom(V, W)</math> sending a vector and a covector to a rank 1 linear transformation ([[simple tensor]] of type (1, 1)), while the inner product is the bilinear evaluation map <math>V^* \times V \to F</math> given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.

The inner product and outer product should not be confused with the [[interior product]] and [[exterior product]], which are instead operations on [[vector field]]s and [[differential form]]s, or more generally on the [[exterior algebra]].

As a further complication, in [[geometric algebra]] the inner product and the {{em|exterior}} (Grassmann) product are combined in the geometric product (the Clifford product in a [[Clifford algebra]]) – the inner product sends two vectors (1-vectors) to a scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called the {{em|outer product}} (alternatively, {{em|[[wedge product]]}}). The inner product is more correctly called a {{em|scalar}} product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).