Page 18 - 35Linear Algebra

18 What is Linear Algebra?

and lines to be explicitly drawn, but we are being diagrammatic; we only
drew four of each.
Now think about adding L(u) and L(v) to get yet another vector L(u) +
L(v) or of multiplying L(u) by c to obtain the vector cL(u), and placing both
on the right blob of the picture above. But wait! Are you certain that these
are possible outputs!?
Here’s the answer

The key to the whole class, from which everything else follows:

1. Additivity:

L(u + v) = L(u) + L(v) .

2. Homogeneity:

L(cu) = cL(u) .

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Most functions of vectors do not obey this requirement. At its heart, linear
algebra is the study of functions that do.

Notice that the additivity requirement says that the function L respects
vector addition: it does not matter if you first add u and v and then input
their sum into L, or first input u and v into L separately and then add the
outputs. The same holds for scalar multiplication–try writing out the scalar
multiplication version of the italicized sentence. When a function of vectors
obeys the additivity and homogeneity properties we say that it is linear (this
is the “linear” of linear algebra). Together, additivity and homogeneity are
called linearity. Are there other, equivalent, names for linear functions? yes.

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E.g.: If f(x) = x then f(1 + 1) = 4 6= f(1) + f(1) = 2. Try any other function you
can think of!

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