0Introduction

IB Linear Algebra

0 Introduction

In IA Vectors and Matrices, we have learnt about vectors (and matrices) in a

rather applied way. A vector was just treated as a “list of numbers” representing

a point in space. We used these to represent lines, conics, planes and many

other geometrical notions. A matrix is treated as a “physical operation” on

vectors that stretches and rotates them. For example, we studied the properties

of rotations, reflections and shears of space. We also used matrices to express

and solve systems of linear equations. We mostly took a practical approach in

the course.

In IB Linear Algebra, we are going to study vectors in an abstract setting.

Instead of treating vectors as “lists of numbers”, we view them as things we

can add and scalar-multiply. We will write down axioms governing how these

operations should behave, just like how we wrote down the axioms of group

theory. Instead of studying matrices as an array of numbers, we instead look at

linear maps between vector spaces abstractly.

In the course, we will, of course, prove that this abstract treatment of linear

algebra is just “the same as” our previous study of “vectors as a list of numbers”.

Indeed, in certain cases, results are much more easily proved by working with

matrices (as an array of numbers) instead of abstract linear maps, and we don’t

shy away from doing so. However, most of the time, looking at these abstractly

will provide a much better fundamental understanding of how things work.