Chapter 4:
Vectors, Matrices, and Linear Algebra
Scott Owen & Greg Corrado
Linear Algebra is strikingly similar to the algebra you learned in high school, except that in the
place of ordinary single numbers, it deals with vectors.
Many of the same algebraic operations you’re
used to performing on ordinary numbers (a.k.a. scalars), such as addition, subtraction and multiplication,
can be generalized to be performed on vectors.
We’ll better start by defining what we mean by
scalars
and
vectors
.
Definition:
A scalar is a number.
Examples of scalars are temperature, distance, speed, or mass – all
quantities that have a magnitude but no “direction”, other than perhaps positive or negative.
Okay, so scalars are what you’re used to.
In fact we could go so far as to describe the algebra you
learned in grade school as
scalar
algebra, or the calculus many of us learned in high school as
scalar
calculus, be cause both dealt almost exclusively with scalars.
This is to be contrasted with
vector
calculus
or
vector
algebra, that most of us either only got in college if at all.
So what
is
a vector?
Definition:
A vector is
a list of numbers
.
There are (at least) two ways to interpret what this list of
numbers mean:
One way to think of the vector as being
a point in a space
.
Then this list of numbers is a
way of identifying that point in space, where each number represents the vector’s component that
dimension.
Another way to think of a vector is
a magnitude and a direction
, e.g. a quantity like velocity
(“the fighter jet’s velocity is 250 mph north-by-northwest”).
In this way of think of it, a vector is a
directed arrow pointing from the origin to the end point given by the list of numbers.
In this class we’ll denote vectors by including a small arrow overtop of the symbol like so:
!
a
.
Another common convention you might encounter in other texts and papers is to denote vectors by use of
a boldface font (
.
An example of a vector is
!
a
=
[4, 3]
.
Graphically, you can think of this vector as an
arrow in the
x-y
plane, pointing from the origin to the point at
x
=3,
y
=4 (see illustration.)
In this example, the list of numbers was only two elements long,
but in principle it could be any length.
The dimensionality of a vector is
the length of the list.
So, our example
!
a
is 2-dimensional because it is a
list of two numbers.
Not surprisingly all 2-dimentional vectors live in a
plane.
A 3-dimensional vector would be a list of three numbers, and they
live in a 3-D volume.
A 27-dimensional vector would be a list of twenty-
seven numbers, and would live in a space only Ilana’s dad could visualize.