Similarly as in the theory of other algebraic structures, linear algebra studies mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F, a linear transformation (also called linear map, linear mapping or linear operator) is a map
that is compatible with addition and scalar multiplication:
for any vectors u,v ∈ V and a scalar a ∈ F. When a bijective linear mapping exists between two vector spaces (that is, every vector from the second space is associated with exactly one in the first), we say that the two spaces are isomorphic. Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view. One essential question in linear algebra is whether a mapping is an isomorphism or not, and this question can be answered by checking if the determinant is nonzero. If a mapping is not an isomorphism, linear algebra is interested in finding its range (or image) and the set of elements that get mapped to zero, called the kernel of the mapping.
Linear transformations have geometric significance. For example, 2 × 2 real matrices denote standard planar mappings that preserve the origin.
Other articles related to "linear transformations, linear, linear transformation, transformations":
... reveal their essential features when related to linear transformations, also known as linear maps ... A real m-by-n matrix A gives rise to a linear transformation Rn → Rm mapping each vector x in Rn to the (matrix) product Ax, which is a vector in Rm ... Conversely, each linear transformation f Rn → Rm arises from a unique m-by-n matrix A explicitly, the (i, j)-entry of A is the ith coordinate of f(ej), where ej = (0...0,1,0 ...
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