From the scalar product, one can define a norm by letting . The scalar product also allows one to define an angle between two vectors: In -dimensional Euclidean space, the standard scalar product (called the dot product) is given by: Cross product in 3-dimensional space 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 determinant: Composition of linear mappings A linear mapping can be defined as a function f between two vector spaces V and W with underlying field F, satisfying If one only considers finite dimensional vector spaces, then in which bV and bW denote the bases of V and W, and vi denotes the component of v on bVi, and 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 Or in matrix form: in which the i-row, j-column element of F, denoted by Fij, is fji, and Gij=gji. The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication. Product of two matrices Given two matrices and their product is given by 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) dimensions of vector spaces U, V and W. Let be a basis of U, be a basis of V and be a basis of W. In terms of this basis, let be the matrix representing f : U → V and be the matrix representing g : V → W. Then is the matrix representing . In other words: the matrix product is the description in coordinates of the composition of linear functions. Tensor product of vector spaces Given two finite dimensional vector spaces V and W, the tensor product of them can be defined as a (2,0)-tensor satisfying: where V* and W* denote the dual spaces of V and W. For infinite-dimensional vector spaces, one also has the: Tensor product of Hilbert spaces Topological