section_path: Matrices
section_name: Matrices
paragraph_text: Matrices
  
  Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is thus an essential part of linear algebra.
  
  Let V be a finite-dimensional vector space over a field F, and (v, v,..., v) be a basis of V (thus m is the dimension of V). By definition of a basis, the map
  
  is a bijection from F, the set of the sequences of m elements of F, onto V. This is an isomorphism of vector spaces, if F is equipped of its standard structure of vector space, where vector addition and scalar multiplication are done component by component.
  
  This isomorphism allows representing a vector by its inverse image under this isomorphism, that is by the coordinate vector (a,..., a) or by the column matrix
  
  If W is another finite dimensional vector space (possibly the same), with a basis (w,..., w), a linear map f from W to V is well defined by its values on the basis elements, that is (f (w),..., f (w)). Thus, f is well represented by the list of the corresponding column matrices. That is, if
  
  for j = 1,..., n, then f is represented by the matrix
  
  with m rows and n columns.
  
  Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing exactly the same concepts.
  
  Two matrices that encode the same linear transformation in different bases are called similar. It can be proved that two matrices are similar if and only if one can transform one into the other by elementary row and column operations. For a matrix representing a linear map from W to V, the row operations correspond to change of bases in V and the column operations correspond to change of bases in W. Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from W to V, there are bases such that a part of the basis of W is mapped bijectively on a part of the basis of V, and that the remaining basis elements of W, if any, are mapped to zero. Gaussian elimination is the basic algorithm for finding these elementary operations, and proving these results.