A linear transformation is a map
vector space V into the vector space W
satisfying two conditions.
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.
be a basis for V and
be a basis 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 W. Then