Page 19 - 35Linear Algebra

1.3 What are Linear Functions? 19

Function = Transformation = Operator

And now for a hint at the power of linear algebra. The questions in
examples (A-D) can all be restated as

Lv = w

where v is an unknown, w a known vector, and L is a known linear transfor-
mation. To check that this is true, one needs to know the rules for adding
vectors (both inputs and outputs) and then check linearity of L. Solving the
equation Lv = w often amounts to solving systems of linear equations, the
skill you will learn in Chapter 2.
A great example is the derivative operator.

Example 4 (The derivative operator is linear)
For any two functions f(x), g(x) and any number c, in calculus you probably learnt
that the derivative operator satisfies

1. d (cf) = c d f,
dx dx
2. d (f + g) = d f + d g.
dx dx dx
If we view functions as vectors with addition given by addition of functions and with
scalar multiplication given by multiplication of functions by constants, then these
familiar properties of derivatives are just the linearity property of linear maps.

Before introducing matrices, notice that for linear maps L we will often
write simply Lu instead of L(u). This is because the linearity property of a

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