linear-guest

12 What is Linear Algebra?
This example is a hint at a much bigger idea central to the text; our choice of
order is an example of choosing a basis 3 .
The main lesson of an introductory linear algebra course is this: you
have considerable freedom in how you organize information about certain
functions, and you can use that freedom to
1. uncover aspects of functions that don’t change with the choice (Ch 12)
2. make calculations maximally easy (Ch 13 and Ch 17)
3. approximate functions of several variables (Ch 17).
Unfortunately, because the subject (at least for those learning it) requires
seemingly arcane and tedious computations involving large arrays of numbers
known as matrices, the key concepts and the wide applicability of linear
algebra are easily missed. So we reiterate,
Linear algebra is the study of vectors and linear functions.
In broad terms, vectors are things you can add and linear functions are
functions of vectors that respect vector addition.
1.2 What are Vectors?
Here are some examples of things that can be added:
Example 2 (Vector Addition)
(A) Numbers: Both 3 and 5 are numbers and so is 3 + 5.
(B) 3-vectors:
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 0 1
⎝1⎠ + ⎝1⎠ = ⎝2⎠.
0 1 1
3 Please note that this is an example of choosing a basis, not a statement of the definition
of the technical term “basis”. You can no more learn the definition of “basis” from this
example than learn the definition of “bird” by seeing a penguin.
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