Many difficult science problems can handled using the powerful, yet easy to use, mathematics of linear algebra. Unfortunately, because the subject (at least for those learning it) requires seemingly arcane and tedious computations involving large arrays of number known as matrices, the key concepts and the wide applicability of linear algebra are easily missed. Therefore, before we equip you with matrix skills, let us give some hints about what linear algebra is. The takeaway message is
Linear algebra is the study of vectors and linear transformations.
In broad terms, vectors are things you can add and linear transformations are very special functions of vectors that respect vector addition. To understand this a little better, lets try some examples. Please be prepared to change the way you think about some familiar mathematical objects and keep a pencil and piece of paper handy!
- 1.1: What Are Vectors?
- Vectors are things you can add and scalar multiply.
- 1.2: What Are Linear Functions?
- In calculus classes, the main subject of investigation was functions and their rates of change. In linear algebra, functions will again be focus of your attention, but now functions of a very special type.
- 1.3: What is a Matrix?
- Matrices are linear functions of a certain kind. One way to learn about them is by studying systems of linear equations.
- Thumbnail: In three-dimensional Euclidean space, these three planes represent solutions of linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. (CC BY-SA 3.0; Alksentrs via Wikipedia)