Friday January 16
−
Lecture 8
:
Linear operators on R
n
.
(Refers to section 3.3
)
Expectations
:
1.
Define linear operator.
2.
Recognize linear operators.
3.
Apply simple linear operators
4.
Show projections are linear operators.
8.1
Definition
−
A linear transformation mapping
R
n
into
R
n
is called a
linear operator
.
8.1.1
Familiar examples of linear operators on
R
2
:
•
Rotation of a vector counterclockwise about
the origin by an angle of
θ .
For a chosen value of
θ
we have the shown that the matrix induced by this linear
transformation is:
⎡
cos
θ
−
sin
θ
⎤
⎣
sin
θ
cos
θ
⎦
•
Reflection transformation about the x-axis
.The matrix which is induced by
this linear transformation was shown to be
⎡
1
0
⎤
⎣
0
−
1
⎦
since
⎡
1
0
⎤
⎡
x
⎤
⎡
x
⎤
⎣
0
−
1
⎦
⎣
y
⎦
=
⎣
−
y
⎦
8.2
We study some more linear operators on
R
2
as well as on
R
3
.
8.2.1
The
projection of a vector
b
onto a vector
a
is given by the expression

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p
= proj
a
(
b
) =
[ <
a
,
b
>
]
(
a
/ ||
a
||
2
) .
8.2.1.1
Example
−
Consider the vector
b
= (
−
4, 3) and let
a
= (2, 0). It is easy, in
this case, to graphically visualize, and then compute what proj
a
(
b
) is.
•
Then proj
a
(
b
) =
−
2(2, 0) = (
−
4, 0) and
w
= (
−
4, 3)
−
(
−
4, 0) = (0, 3).
8.2.1.2
(
This proof is optional material although one is expected to understand
the flow of it. However, one is expected to know by heart the formula for
proj
a
(
b
).)
Proof of a formula for the projection
p
= proj
a
(
b
) of
b
onto
a
. (
Optional
)
•
By the definition the points
b
,
p
=
t
a
and
w = b
−
p
form a right triangle in a
plane (
b
forming the hypotenuse), so <
p
,
b
−
p
> = 0.
•
Recall that
a
/ ||
a
|| is the unit vector in the direction of
a
. Since
p
=
t
a
,
then
a
/ ||
a
||
=
±
p
/ ||
p
|| .

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