Most students entering this course will have experience with high school mathematics and first year calculus (and maybe a smattering of other experienes, such as statistics). Linear algebra is not only a new topic in mathematics; compared to these previous experiences, it involves a substantially new way of thinking. In this introduction, I want to briefly lay out the main themes and ideas that I’ll be working through for the entire term. I encourage you to keep these ideas in the back of your head and notice how the material of the course builds up these theme.
Geometry and Algebra. Much of modern mathematics is based on the cartesian coordinate system. The major strength of this system is that is builds a connection between geometry and algebra. Shapes are connected to their equations, so that you want determine the properties of shapes by working with those equations. Likewise, equations can be interpreted as shapes, so that you can use geometry to visualize what is happening in algebra. In calculus, this happened through graphs: a graph was a geometric visualization of the behaviour of a function. Linear algebra continues to build on this connection, though the specific types of geometry and equations are different from calculus. All of the equations and algebraic manipulation in the course can be visualized geometrically and, likewise, all of the geometry in the course can be accessed algebraically. Though it is called an ‘algebra’ course, this is as much a geometry course as anything.
Flatness. If linear algebra is a geometry course, what kind of geometry does it do? It does flat geometry: straight lines, planes, flat surfaces, and higher dimensional analogues. Flat objects are inherently more accessible and understandable than more complicated curved objects, so they are a natural start to geometry. Much of more complicated geometry is built on the linear algebra foundation of flat geometry. It’s a natural place to start.
Linearity. In geometry, the word ‘linearity’ refers to lines, which fits with the previous point about flat objects. However, the word also has another important meaning in algebra: an operation or structure is linear if it interacts well with addition, subtraction and multiplication by constants. That’s a bit vague for now, but I’ll be developing this idea in detail over the course. From the algebriac side, this course is about understanding how these linear operations behave and what they can and cannot do. And, of course, the geometric linear and the algebraic linear do match up with each other.
Symmetry. In ordinary usage, a symmetry is a change of a shape that preserves it, like a reflection. The mathematical notion is similar, but not precisely the same. If I have a transformation (such as a reflection), its symmetries are the things its preserves. I’ve just flipped the definition here: in conventional usage, the shape has a symmetry, but in mathematical usage, the transformation (the reflection) has the symmetry. Mathematically, whenever I define some kind of transformation, I want to ask for its symmetries: for what is preserves. If can preserve shapes, and I will talk about that in detail. But it can also preserve other mathematical structures, particularly algebraic structures. to study symmetry is to ask, for a mathematical operation: what is preserved? Asking this quesiton can build a deep understanding of the transformation. I’ll keep coming back to this question of what is perserved throughout the course.
Definition, Rigor and Proof. Mathematics is a formal language, with strict rules about logic, definition and arguments. Your mathematical education to date has included this formalization, but this course represents a substantial step up in commitment to that formalization. All the definitions in the course will be given in mathematically formal language. I will try to provide proofs for many of the statements. I’ll also ask you to try to write proofs in your exercises and assignments. I’ll try to convince you that writing proofs, along side doing calculations or solving problems, is ‘doing mathematics’.
Subsection1.1.2Mathematical Thinking and Writing
The themes listed above are the content goals I have for the course. However, in addition to the content of linear algebra itself, I want to focus on the more general skills of mathematical thinking and writing. I’m going to present material more formally and expect a higher standard of formal writing from you. A large part of learning mathematics is being apprenticed into the way that mathematics is written and, evern more so, the way that mathematics is thought. This is a much trickier task than just learning how to do problems: it’s learning a style, not an algorithm. It’s learning a way of thinking, not just an individual fact. As challening as it is, working on thinking and writing is a major goal of this course. I hope you buy into this challenge, that you consciously work on your mathematical writing and thinking.
Of the undergraduate mathematics courses, linear algebra is the most natural place to start talking about mathematical proofs and to work on formalized mathematical writing. The material lends itself to a number of accessible proof examples and have a level of abstraction that suits proof questions. The linear algebra course also serves as a prerequisite for most of the other mathematic courses where proof becomes quite important, including the senior calculus courses, discrete mathematics, abstract algebra, cryptography and theory of computing.
