This course is an introduction to linear algebra. It has been argued that linear algebra constitutes half of all mathematics.
Whether or not everyone would agree with that, it is certainly true that practically every modern technology relies on linear algebra to simplify the computations required for Internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between.
Linear algebra can be viewed either as the study of linear equations or as the study of vectors. It is tied to analytic geometry; practically speaking, this means that almost every fact you will learn in this course has a picture associated with it. Learning to connect the facts with their geometric interpretation will be very useful for you.
The book which is used in the course focuses both on the theoretical aspects as well as the applied aspects of linear algebra. As a result, you will be able to learn the geometric interpretations of many of the algebraic concepts in this subject. Additionally, you will learn some standard techniques in numerical linear algebra, which allow you to deal with matrices that might show up in applications. Toward the end, the more abstract notions of vector spaces and linear transformations on vector spaces will be introduced.
In college algebra, one becomes familiar with the equation of a line in two-dimensional space: y = mx+b. Lines can be generalized to planes and “hyperplanes” in many-dimensional space; these objects are all described by linear relations. Linear transformations are ways of rotating, dilating, or otherwise modifying the underlying space so that these linear objects are not deformed. Linear algebra, then, is the theory and practice of analyzing linear relations and their behavior under linear transformations. According to the second interpretation listed above, linear algebra focuses on vectors, which are mathematical objects in many-dimensional space characterized by magnitude and direction. You can also think of them as a string of coordinates. Each string may represent the state of all the stocks traded in the DOW, the position of a satellite, or some other piece of data with multiple components. Linear transformations change the magnitude and direction of vectors—they transform the coordinates without changing their fundamental relationships with one another. Linear transformations are often written in a compact and easily-readable way by using matrices.
Linear algebra may at first seem dry and difficult to visualize, but it is one of the most useful subjects you can learn if you wish to become a business-person, a physicist, a computer-programmer, an engineer, or a mathematician.
