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Lecture 29

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Mathematics

MATH 125

Scott

Fall

Description

Recap
Let T: Rn Rp be a linear transformation.
The standard matrix of T is the p x n matrix
For any UE Rn, we have
If S: Rp +Rm is another linear transformation, then the
composition So
T: R" RIT is defined by setting
This is again a linear transformation.
Solution
Theorem 6.12: Let T: Rn Rp and S Rp Rm be
linear transformations. Then the standard matrix of the com
position SoT is equal to the product of the standard matrices
of the individual transformations:
Proof
Let v e R n. Then
Remark: More generally, if Ti, T2, TR are linear trans-
formations for which the composition Tk o Tr
o To o Th is
defined, then
o T2 o T
T. T
Example 6.13: Let T: R2 R4 and S: R4 Ra be the
linear transformations given by
T2
and S
4r
T4
Recap Let T: Rn Rp be a linear transformation. The standard matrix of T is the p x n matrix For any UE Rn, we have If S: Rp +Rm is another linear transformation, then the composition So T: R" RIT is defined by setting This is again a linear transformation. Solution Theorem 6.12: Let T: Rn Rp and S Rp Rm be linear transformations. Then the standard matrix of the com position SoT is equal to the product of the standard matrices of the individual transformations: Proof Let v e R n. Then Remark: More generally, if Ti, T2, TR are linear trans- formations for which the composition Tk o Tr o To o Th is defined, then o T2 o T T. T Example 6.13: Let T: R2 R4 and S: R4 Ra be the linear transformations given by T2 and S 4r T4Find the standard matrix IS o T of the composition S o T
using
(1) the definition, and
(2) matrix multiplication.
Solution
(1) Find (s oT)CET) an
(s. T) EL')
(2) In
CT1 and [S]
1
Fina
O 3
1 4
1 1 1
0 1
4 0
1 -A
o 1 -1
O 2
Example 6.14: Find the standard matrix of the linear oper-
ator T: R2 IR2 that performs a counter-clockwise rotation
about the origin through radians followed by reflection about
the y-axis.
Solution:
(TT/2) -sin( /2)
otation T1)
2) cos /2
Reflection (T2)
Find the standard matrix IS o T of the composition S o T using (1) the definition, and (2) matrix multiplication. Solution (1) Find (s oT)CET) an (s. T) EL') (2) In CT1 and [S] 1 Fina O 3 1 4 1 1 1 0 1 4 0 1 -A o 1 -1 O 2 Example 6.14: Find the standard matrix of the linear oper- ator T: R2 IR2 that performs a counter-clockwise rotation about the origin through radians followed by reflection about the y-axis. Solution: (TT/2) -sin( /2) otation T1) 2) cos /2 Reflection (T2)n about tu
line y x
reflecti
Invertibilility of linear operators
Definition: The linear operator In: Rn Rn defined by
setting
In(i)
for all i E R" is called the identity map on R
Since In
e for all 1 S is n, the standard matrix of L
is the n x n identity matrix In.
([In 3In)
Definition: Let T: R" Rn be a linear operator on Rn
We say that T is invertible if there exists another linear
operator S: R" R" such that
S o T and T o S I
In this case we say that S is an inverse of T
Note: As with square matrices, if T is invertible, then it has
a unique inverse, and we denote it by T
Example 6.15: The linear operator T: R2 R2 that re-
ffects vectors about a line l through the origin is invertible. In
fact, T
T Cra Plechng twice
is the identit
Example 6.16: The linear operator T: R2 R2 given by
counter-clockwise rotation about the origin through 0 radians
is invertible. The inverse operator T
1 is given by clock
Wisc.
tation about the origin through radians
The concept of invertibility for linear operators on R" is in fact
equivalent to the concept of invertibility for n x n matrices:
Theorem 6.17: A linear operator T: Rn R" is invertible
if and only if its standard matrix Tj s invertible. Moreover,
in this case, we have
Example 6.18: The standard matrix of the linear operator
T on R2 given by reflection about the z-axis is
(Example 6.8)
Since T
1 T (Example 6.15), we have T
n about tu line y x reflecti Invertibilility of linear operators Definition: The linear operator In: Rn Rn defined by setting In(i) for all i E R" is called the identity map on R Since In e for all 1 S is n, the standard matrix of L is the n x n identity matrix In. ([In 3In) Definition: Let T: R" Rn be a linear operator on Rn We say that T is invertible if the

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