Page 17 - 35Linear Algebra

1.3 What are Linear Functions? 17

While this sounds complicated, linear algebra is the study of simple func-
tions of vectors; its time to describe the essential characteristics of linear
functions.
Let’s use the letter L to denote an arbitrary linear function and think
again about vector addition and scalar multiplication. Also, suppose that v
and u are vectors and c is a number. Since L is a function from vectors to
vectors, if we input u into L, the output L(u) will also be some sort of vector.
The same goes for L(v). (And remember, our input and output vectors might
be something other than stacks of numbers!) Because vectors are things that
can be added and scalar multiplied, u + v and cu are also vectors, and so
they can be used as inputs. The essential characteristic of linear functions is
what can be said about L(u + v) and L(cu) in terms of L(u) and L(v).
Before we tell you this essential characteristic, ruminate on this picture.

The “blob” on the left represents all the vectors that you are allowed to
input into the function L, the blob on the right denotes the possible outputs,
and the lines tell you which inputs are turned into which outputs. 6 A full
pictorial description of the functions would require all inputs and outputs

6
The domain, codomain, and rule of correspondence of the function are represented by
the left blog, right blob, and arrows, respectively.

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