Page 112 - 35Linear Algebra

112 Linear Transformations

6.1 The Consequence of Linearity

Now that we have a sufficiently general notion of vector space it is time to
talk about why linear operators are so special. Think about what is required
to fully specify a real function of one variable. One output must be specified
for each input. That is an infinite amount of information.
By contrast, even though a linear function can have infinitely many ele-
ments in its domain, it is specified by a very small amount of information.

Example 69 (One output specifies infinitely many)
If you know that the function L is linear and that

1 5
L =
0 3
then you do not need any more information to figure out

2 3 4 5
L , L , L , L , etc . . . ,
0 0 0 0
because by homogeneity

5 1 1 5 25
L = L 5 = 5L = 5 = .
0 0 0 3 15
In this way an an infinite number of outputs is specified by just one.

2
Example 70 (Two outputs in R specifies all outputs)
Likewise, if you know that L is linear and that

1 5 0 2
L = and L =
0 3 1 2
then you don’t need any more information to compute

1
L
1
because by additivity

1 1 0 1 0 5 2 7
L = L + = L + L = + = .
1 0 1 0 1 3 2 5
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