Linear Algebra and Geometry

Let’s have a short lecture about geometry more often than not mathematics works based on analogies with

linear algebra that’s extremely relevant.

We said that a scalar has no dimensions.

What else has no dimensions that you can think of.

Well the geometrical idea of a point a point is obviously the simplest object in terms of geometry.

It has no direction nor size.

A scalar is like a point.

Then we have vectors.

A vector has one dimension.

It’s like a line.

However it has a direction.

It can be oriented in many ways.

I would really like to plot this on a two dimensional plane however to get into 2-D.

We need something else.

The only way to get from a line to a plane is by crossing it with another line.

Right.

So two dimensional space is defined by two lines.

Well two lines means two vectors and in linear algebra a matrix is a collection of vectors.

Therefore any two dimensional plane can be represented by a matrix this idea is very simple yet extremely

powerful.

Let’s explore the two dimensional plane.

If I want to represent a vector say to 5 it would be a line from the origin of the plane to the point

of position 2 5.

The direction is always from the origin of the graph to the endpoint the vector minus two minus five

would be exactly the opposite of the first one.

The vector minus 2 3 would appear the following way.

And so on as you can see each of these vectors has a direction determined by its values.

OK.

By the way this vector is 1 0 while the other one 0 1.

Taking these two vectors together we can create a matrix 1 0 0 1 this matrix is basically made up of

small portions of the two axes x and y.

What’s important though is that if I remove everything else from the graph this matrix would define

the two dimensional plane we were just looking at actually every two vectors I take would define the

2d plane.

We are looking at in fact are you watching the course.

All these major C’s are defining the plane of your screen right now.

Cool.

A nice free online tool to play around with is the 3D vector plotter Bayaka the morgue.

You can specify each vector of interest and fiddle with the plots.

This Web site allows you to create all kinds of matrices up to three by three.

In this way you will get a better grasp of the geometrical concept accompanying vectors and matrices.

Great let’s get out of geometry and back to linear algebra.

Note that you can come back to this lecture at any future point and look into these ideas with fresh

eyes.

Thanks for watching.