Another way to express this is that any vector space is completely classified (up to isomorphism) by its dimension, a single number. In particular, any n-dimensional -vector space is isomorphic to . There is, however, no "canonical" or preferred isomorphism; actually an isomorphism is equivalent to the choice of a basis of , by mapping the standard basis of to , via . The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context; see below. Matrices Matrices are a useful notion to encode linear maps. They are written as a rectangular array of scalars as in the image at the right. Any -by- matrix gives rise to a linear map from to , by the following where denotes summation, or, using the matrix multiplication of the matrix with the coordinate vector Moreover, after choosing bases of and , any linear map is uniquely represented by a matrix via this assignment. The determinant of a square matrix is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero. The linear transformation of corresponding to a real n-by-n matrix is orientation preserving if and only if its determinant is positive. Eigenvalues and eigenvectors Endomorphisms, linear maps , are particularly important since in this case vectors can be compared with their image under , . Any nonzero vector satisfying , where is a scalar, is called an eigenvector of with eigenvalue . Equivalently, is an element of the kernel of the difference (where Id is the identity map . If is finite-dimensional, this can be rephrased using determinants: having eigenvalue is equivalent to By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in , called the characteristic polynomial of . If the field is large enough to contain a zero of this polynomial (which automatically happens for algebraically closed, such as ) any linear map has at least one eigenvector. The vector space may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map. The set of all eigenvectors corresponding to a particular eigenvalue of forms a vector space known as the eigenspace corresponding to the eigenvalue (and ) in question. To achieve the spectral theorem,