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MAT211 Lecture 13
The matrix of a linear transformation
✦
The Bmatrix of a linear transformation
✦
The columns of the Bmatrix of a linear transformation
✦
Change of basis matrix
✦
Change of basis in a subspace of R
n
✦
Change of basis for the matrix of a linear transformation
A
and
B
basis of linear space V,
T a linear transformation from V to
V.
•
Coordinate Transformation from V to R
n
L(f)=
[f]
B
•
B
matrix of T is
L
B
o T o L
B
1
•
Change of basis from B to A, S
B>A
=L
A
o (L
B
)
1
•
If B is
B
matrix of T and A is
A
matrix of T,
S the change of basis from B to A,
AS=S B
Overview
3
EXAMPLE
•
Consider the space U of upper triangular
2 x 2 matrices and the linear
transformation T from U to U deFned by
T(M)=AM where A is
1
2
0
3
±or each element z of U,
Fnd [T(z)]
B,
where B is
the standard basis
4
DeFnition
•
Consider a linear transformation T from V
to V where V is an ndimensional linear
space. Let
B
denote a basis of
V.
•
The matrix B of the transformation from R
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 Fall '08
 MOVSHEV

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