What is linear algebra?
This section is (very) non-rigorous, and aims to provide some sort of intuition on the answer to this question without delving into nightmarish abstractions.
The question has two components: what is the meaning of the words linear and algebra in this context?
What is linear?
Exercise (basic): Recall graphs of lines
Recall graphs of linear functions: choose any
and and recall why does the graph of look like a line. You might want to start with plugging in various values of and, in an experimental fashion, see that the points on the graph always land on one single line. You could try to think of a general reasoning that would explain this experimental observation.
Exercise: Draw a graph of a linear function of several variables
Consider a function
. Can you draw the graph of this function? *Careful! This graph will be in -dimensional space: two dimensions for the variables and and one dimension for the value of the function. Linear = behaving linearly, like a line. We might remember this word from school, when we studied linear functions . We saw that their graphs looked like lines on the plane, hence the name. However, this meaning of linear can be slightly misleading in our scope.
Let us imagine that we live on the surface of a sphere. We are flat creatures in this setting. Imagine that you are moving along the sphere. What would describe the direction of your movement at any given moment of time? A vector tangent to the sphere. What would describe the entire set of your possibilities to move from one given point? The tangent plane to the sphere at your standing point. If you turn, how does the set of your possibilities to move change? The tangent plane rotates around its origin, staying in place. This rotation is an example of something we will be calling a linear map.
In other words, the set of directions of (instantaneous) movement is something “linear”.
Another, more directly practical, motivation for thinking about linearity is the following: imagine you have two number parameters of some experiment/dataset. For example, imagine you are investigating the determining factors of a certain disease, and, among the data you collect from patients, you collect weight and height. Your task is to try to predict which weight and height combination could indicate a higher risk of the disease presence. You can plot this data on the plane, and, very often, you see that it seems reasonably close to a line. The actual dependence (you can argue that one doesn’t exist, even) might be very complicated, but, in a lot of realistic scenarios, even constructing a line gives very good prediction results. Essentially, the simplest possible dependence between quantities is often the correct one. For this reason, linear algebra is omnipresent in data analysis and statistics: you want to be able to efficiently manipulate linear stuff.
What is algebra?
Sidenote: Sets
We have used the word “set” several times without really precising what we mean by it. Although it would seem that a set is something trivial to define as an ensemble of elements (where elements can be anything), there are very non-trivial complications with a naive definition. They are so non-trivial that they have only been discovered in the beginning of the 20-th century, and, for quite a bit of time, “broke” mathematics. Thankfully, we will never have to deal with these arising difficulties due to the “niceness” of sets in linear algebra, but there will be some weird examples of this behavior later in the course.
Contrarily to what you might be used from school, where algebra usually refers to equations, inequalities, variables etc., in research mathematics the word algebra usually translates to working with structures. Very loosely speaking, algebra is about formalizing what kind of properties follow from the existence of operations, orders and so on, on sets. For example, we know that the real numbers have two operations, addition and multiplication, that have a lot of nice properties and are intertwined by the distributive property
So, what is linear algebra?
The standard definition would be: “Linear algebra is the study of linear maps of vector spaces.” Translating to our language on the sphere, we try to establish what happens (from our perspective) with our direction set when we shrink, rotate or move ourselves, and this describes the “linear maps” part of the definition. What about the “vector spaces”? This is the algebraic part: we will investigate the question using a language based on a very general definition (based on examples that will provide us some geometric intuition behind the symbols) of a vector space. Remember the example with real numbers? We strip the usual vectors of their attachment to real numbers, and introduce an abstract object with the properties of usual vectors, but not necessarily vectors themselves (as we will see in examples motivating the definition).
Vector space — the main abstraction of linear algebra
Transclude of Vector-space-(over-real-numbers)#^c30824
This definition does not make any sense, does it? :)
This is a work in progress.