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Linear Transformation

Alternate name

linear map

Basic definition

A function from one vector space to another. If bases are chosen for the vector spaces, a linear transformation can be given by a matrix.

Detailed definition

A linear transformation between two vector spaces V and W is a map T:V->W such that the following hold: 1.T(v_1 + v_2) = T(v_1) + T(v_2) for any vectors v_1 and v_2 in V, and 2.T(α v) = α T(v) for any scalar α. A linear transformation may or may not be injective or surjective. When V and W have the same dimension, it is possible for T to be invertible, meaning there exists a T^(-1) such that T T^(-1) = I. It is always the case that T(0) = 0. Also, a linear transformation always maps lines to lines (or to zero).

Related terms

elliptic fixed point | general linear group | hyperbolic fixed point | invertible linear map | involutory | linear algebra | linear operator | matrix | matrix multiplication | parabolic fixed point | vector basis | vector space

Educational grade level

college level

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