LINEAR ALGEBRA (Math 211)
Sheet #7
LINEAR TRANSFORMATIONS

A linear transformation is a map from the vector space V into the vector space W satisfying two conditions.
, and

for any , from V and any from .

Examples. V=W=Pn is the space of polynomials of degree .
(1) .                 (2)Sifting operator:     .

Matrix of a linear transformation T. Let , , ..., be a basis for V and , , ..., be a basis for W. If

then the matrix AT= is called the matrix of the linear transformation T with respect to the chosen bases , ..., for V and , ..., for W.

Theorem. For a vector from V the coordinates of the vector with respect to the basis , ..., can be written as

Composition of linear transformations. Let and be linear transformations. Their composition is a linear transformation from V to U such that . Then , where

 AL is the matrix of L with respect to the bases ,..., for V and , ..., for W; AT is the matrix of L with respect to the bases ,..., for W and , ..., for U; is the matrix of with respect to the bases ,..., for V and , ..., for U.

Transition matrix P from an old basis to a new one .

Theorem. Let AT be the matrix of the linear transformation with respect to the bases ,..., for V and , ..., for W. Then

is the matrix of T with respect to the new bases ,..., for V and , ..., for W, where
PV is the transition matrix from the basis ,..., to the basis ,..., and
PW is the transition matrix from the basis ,..., to the basis ,..., .