Transcript Lecture No. 3

PLANE ALGEBRA
No, it’s not a spelling error, I mean plane, not
plain. Plain algebra is:
or
.
We are going to do “algebra of the plane.”
We begin, as usual, by fixing some terminology.
The plane we will call
because, thanks to
René Descartes, we can think of it as
ordered pairs of real numbers
(what does ordered mean?).
That’s right, it means that order matters, so
the pair
is distinct from the pair
.
For later purposes of useful arrangement, we
write the two real numbers vertically
(as a 2x1 matrix)
and call it a vector.
Please do NOT read any additional meaning into
the word vector, other than
a vertical arrangement (two rows, one column)
of two real numbers.
Pictorial and physical interpretations of these 2x1
matrices called vectors will come later, we are
playing with algebra, not geometry (for now!)
What can we reasonably do with these pairs of
numbers? Here are the usual symbols for algebraic
operations:
With the knowledge acquired through the
experience of past mathematicians, we know that
the
and
can be easily extended, the
can also be extended, with limitations, and for
now we’ll forget about
.
The appropriate definitions are:
Definition. (The and the ). Let
be any two vectors. We define
to be the vector
(The
)
For any real number
we define
and any vector
to be the vector
and
.
Some remarks are important here:
1. Sums and differences of vectors are vectors.
2. A real number times a vector is a vector.
In this context (remember our discussion?)
the real number we multiply by is called a
scalar.
3. There is nothing to limit ourselves to vectors
consisting of two real numbers, we most
certainly could use three real numbers (we
will call that collection of vectors
) or
four real numbers (
) or even n numbers
(
! ) (See p. 27 of your textbook)
From now on, at least in our undetermined
discussions (the undetermined is how many
numbers comprise a vector, the n we mentioned before), we will use lower-case, end-ofthe-alphabet symbols like v, w to represent
vectors.
One of the most important concepts about
vectors is that of a linear combination .
The definition is important enough that we give it
in the next slide.
Definition. Let v1 ,v2 ,v3 ,...,vk be k vectors. A
linear combination of v1 ,v2 ,v3 ,..., vk
is any vector v that can be written as
(in short-hand sigma notation …. ready?
)
for some k scalars c1 ,c2 ,c3 ,..., ck .
Let’s do some examples. We’ll take three vectors
in
and write a couple of combinations.
OK, the examples we just did show that it’s child’s
play to construct vectors that are linear combinations of a given set of vectors. BUT …
Given k vectors v1 ,v2 ,v3 ,...,vk and given
another vector v , can we decide whether or not
the given vector v is or is not a linear combination of the k vectors v1 ,v2 ,v3 ,...,vk ?
Let’s work with
for a while. Here are three
vectors in
(also called 3-dimensional vectors)
and here is another three-dimensional vector
Question: can v be written as a
linear combination of v1 ,v2 ,v3 ?
Let’s see … we need to see if we can find three
numbers (scalars) who1 , who2 , who3 so that
Applying our definitions of scalar multiplication
and vector addition we get the “equation”
(OK, I’ll drop the silliness, who becomes x .)
that we would like to solve for x1 , x2 , x 3 .
Whoa, this is just the linear system
I have
a nice package I can
use to solve it!
(There should be infinitely many solutions!)
Time to look at some pictures. Thanks to René
Descartes we know that
two-dimensional vectors are just points in the
plane, in
. (More to come later)