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*=*P*_{n} is the space of polynomials of degree
.

(1) .
(2)*Sifting operator:*
.

**Matrix of a linear transformation T.**
Let
,
,
...,
be a basis for

then the matrix

**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

A_{L} |
is the matrix of L with respect to the bases
,...,
for V and
,
...,
for W; |

A_{T} |
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 A_{T} be the matrix of the linear transformation
with respect to the bases
,...,
for V
and
,
...,
for W. Then
*