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What Is a Linear Transformation? 171
We can demonstrate the same thing visuay.
We’ use the
2×2 matrix as f.
We’ll show that the first rule holds:
x
2
x
1
x
2j
x
2i
x
1i
x
1j
O
R
2
x
2
x
1
a
11
(x
1i
+ x
1j
) + a
12
(x
2i
+ x
2j
)
a
21
(x
1i
+ x
1j
) + a
22
(x
2i
+ x
2j
)
O
R
2
x
2
x
1
x
2i
+ x
2j
x
1i
+ x
1j
O
R
2
x
2
x
1
a
11
x
1i
+ a
12
x
2i
a
21
x
1i
+ a
22
x
2i
a
11
x
1j
+ a
12
x
2j
a
21
x
1j
+ a
22
x
2j
O
R
2
If you
multiply
first...
then
multiply...
If you
a first...
then
a...
You get the same final result!
a
11
a
21
a
12
a
22
x
1i
x
2i
x
1j
x
2j
+ =
x
1i
+ x
1j
x
2i
+ x
2j
a
11
a
21
a
12
a
22
a
11
a
21
a
12
a
22
a
11
a
21
a
12
a
22
172 Chapter 7 Linear Transformations
If you
multiply
by the
matrix
first...
then
multiply
by the
matrix...
If you multiply
by
c
first...
then multiply
by
c
...
x
2
x
1
x
2i
x
1i
O
R
2
x
2
x
1
a
11
x
1i
+ a
12
x
2i
a
21
x
1i
+ a
22
x
2i
O
R
2
x
2
x
1
cx
2i
cx
1i
O
R
2
x
2
x
1
a
11
(cx
1i
) + a
12
(cx
2i
)
a
21
(cx
1i
) + a
22
(cx
2i
)
O
R
2
You get the same final result!
c
And the second rule, too: c =
x
1i
x
2i
a
11
a
21
a
12
a
22
x
1i
x
2i
a
11
a
21
a
12
a
22
Awesome!

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