Math 112: Linear Algebra

Overview of the Course

The subject of this course is linear algebra.

Linear means the simplest kind of functions (after constants).

For example:

• f(x)=x
• f(x)=3x+1
• f(x)=-5x+1
• f(x)=1/2x-1

These are all linear.

BUT the following:

• f(x)=x^2
• f(x)=sin(x)
• f(x)=x^3-x

The difference is more apparent if we draw graphs:

Linear:

Not Linear:

The linear functions are simpler because they are just straight lines.

In a calculus course, you'll see more complicated functions, and use calculus to analyze them.

We're not going to look at anything more complicated than linear functions. But we are going to look at more than one at once, so things get complicated in a different way.

Eg.

We could just solve it and stop there.

But there are several other ways of looking at the situation:

As a...

• Linear system
• Graph - use the geometry
• Augmented matrix (systemized method of solving.)
• Vector Equation
• Matrix Equation
• Transformation

Looking at something from a different point of view may give a new insight and enable us to solve the problem more easily.

The different ways of looking at the situation:

In this course we're going to look at all the above ideas in some detail.

Outline of contents of lectures

We'll cover the following (not necessarily in exactly the following order). (I shall add corresponding section numbers in the book very soon).

Linear Systems

Definitions, simple examples, notation

The Geometry

How a solution can be represented by a point on a graph. Equations are lines. Lines can meet in different ways.

Transformations

How to think about a linear system as a transformation, and lots of other interesting transformations. Lots of pictures!

Vector spaces

Some interesting properties of solutions of linear systems. We'll see problems with many solutions. "vector spaces" is the name for a certain collection of solutions. We'll see how to get many solutions from a few "basic" solutions. We'll also study dimension. Eg., what is 3 about the 3-dimensional space what we live in? We'll also define inner product.

Vector Equations

Now we know about vector spaces, we'll look at linear systems as vector equations

Matrices

We'll go back and reformulate our ideas about linear systems and transformations in terms of a new notation: Matrices. But matrices are more than just a notation; we can do a lot of interesting things with them in their own right. We'll briefly look at how several other concepts can be formulated in terms of matrices.

Matrix Topics

• Transformations
• Determinant
• Rank
• Matrix multiplication
• Inverse of a matrix
• Eigen vectors, eigen values
• Diagonalization

Applications...

Finale

As a finale, we'll use the ideas in the course to look at some interesting mathematics (not decided what yet, maybe something to do with fractals...)(non examinable)

Please send corrections to spelling mistakes, and any other comments, and ideas about linear algebra, to me.