VECTOR FIGURESHigh time to introduce some visualization in our theory. We start with visualizing

The following figure 1 is kind of universal, even though

we spent a good ten minutes of lecture time to figure

out precisely how it is built.

In fact we did figure out that the next figure 2 is just as good, especially for spaceship traveling aliens.

Figure 4 however will be new to most of us and will require come discussion.

Here is our old friend

figure 1

Next is a recently met friend, when we realized that we could choose any two perpendicular lines for our axes.

Question: which axes is which?Ans.:

As a matter of fact we will continue dropping the and notation and use #1 and #2 instead, this way:

The first thing we want to do is interpret visually the two algebraic operations we have defined on vectors, and scalar multiplication.The easiest is scalar multiplication. We will do that one first.From the theory of

we get that, if and

then the following figure holds:

(remember High School geometry?)

The pointsare collinear. This is the visual meaning of scalar multiplication,a stretching (or compressing !) on the same line.

Actually I should have made eight figures (you wish!) to illustrate the eight cases

You do it!

a b

+ + ++ + -+ - ++ - -- + +- + -- - +- - -

Next we look at .

Let

Then

Once more, with the aid of we get the figure

(Note the color coding)

You can easily verify that

(known as the “parallelogram law.”)

What we have achieved is a visual representation of both vector sum and scalar product in .Let’s return to general theory for awhile. We want to give a name to the set of all the vectors we can generate starting with some fixed ones.More precisely: let be given vectors. Let’s compute all possible linear combinations of these given vectors, that is

for all possible choices of the .What will we get? If the answer is easy.

For we start by giving a name and a symbol to the collection of vectors we get.The name isAnd the symbol isSo we have the equation

In the span of two non-collinear vectors is

all of

as the figure in the next slide shows.(see also Example 4, p. 28 of the textbook.)

Given the two non-collinear vectors ,any other vector can be written as a linear combination of them.

Figure 4

Guesstimate:

In the situation is similar but slightly more complicated.Obviously is just The figure

Shows that is What about

Intuitively we want to say that if the three vec-tors are not co-planar they will “span” all of .But how do you determine if the three vectors

are co-planar? That research comes next, we need some better notation first.

Matrix Equations

We have looked before at an ugly animal like this:

And we shortened it some, to

We will do much better, we will boil down to

Where and are the vectors shown in the

next slide

Ready?

We are going to learn how to handle the coefficient matrix, and the unknown column vector, and the constant terms vector as one unit each, in the next lecture.