Linear transformation

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A linear transformation (also called linear operator or linear map) is a function between two vector spaces which respects the arithmetical operations addition and scalar multiplication defined on vector spaces.

Formally, if V and W are vector spaces over the same ground field K, we say that f : V -> W is a linear transformation if

  • f(x + y) = f(x) + f(y) for all x, y in V
  • f(ax) = a f(x) for all a in K and x in V

From this defintion it follows directly that f(0) = 0 and f(-x) = -f(x).

If V and W are finite dimensional and bases have been chosen, then every linear transformation from V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear transformations: if A is a real m-by-n matrix, then the rule f(x) = Ax describes a linear transformation Rn -> Rm (see Euclidean space).

There are also important examples of linear transformation involving infinite-dimensional spaces. For instance, the integral yields a linear map from the space of all real-valued integrable functions on the interval [a, b] to R, while differentiation is a linear transformation from the space of all differentiable functions to the space of all functions.

The composition of linear transformations is linear: if f : V -> W and g : W -> Z are linear, then so is g o f : V -> Z. In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the multiplication of matrices.

If f : V -> W is linear, we define the kernel and the image of f by

ker(f) = { x in V : f(x) = 0 }
im(f) = { f(x) : x in V }

Ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula is often useful:

dim(ker(f)) + dim(im(f)) = dim(V)

Occasionly, V and W can be considered as vector spaces over different ground fields, and it is then important to specify which field was used for the definition of "linear". If V and W are considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map C -> C, but it is not C-linear.