section_path: Vector spaces/Linear maps
section_name: Linear maps
paragraph_text: Linear maps
  
  Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F, a linear map (also called, in some contexts, linear transformation or linear mapping) is a map
  
  that is compatible with addition and scalar multiplication, that is
  
  for any vectors u, v in V and scalar a in F.
  
  This implies that for any vectors u, v in V and scalars a, b in F, one has
  
  When V = W are the same vector space, a linear map T : V → V is also known as a linear operator on V.
  
  A bijective linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an isomorphism. Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in the sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm.