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.