spring 2003 course description
spring 2003 syllabus
project guidelines
Mathematica files
spring 2003 class profile
course evaluation form
Linear algebra, as its name might suggest, is the part of algebra that considers sets with linear structures and functions that preserve that structure. The most familiar linear structure to most students is the real numbers system: real numbers can be added and multiplied by other real numbers. The only linear functions on the reals are those of the form f(x) = mx, where m is a constant called the slope. The properties that make a function linear are these:
| Linear functions | ||
| f(cx) = cf(x), where c is any "scalar" (number) | ||
| f(x+y) = f(x) + f(y), where x and y are any set elements | ||
You can easily check that functions of the form f(x) = mx satisfy these properties. It is also true that their graphs are lines, the main reason we call them linear. But other linear graphs, such as f(x) = x+1, do not satisfy the two properties above. These functions are called "affine linear": linear with a shift.
Another linear structure is the set of all 2-dimensional vectors in the plane. Vectors can be added, and they can be multiplied by scalars. The only linear functions on vectors are represented by 2 × 2 matrices. This family of functions is fairly varied - they can stretch, skew, reflect, and rotate. Matrix functions are useful for computer-generated graphic effects. The background on this webpage is made with four affine linear functions.
Other, more exotic, linear structures include Euclidean n-space (n as large as you like), infinite-dimensional spaces, the set of solutions to equations of various sorts, the set of all square-summable real or complex sequences, certain sets of functions, and certain sets of matrices. We will meet some of these; others await your arrival in later math courses.
The study of linear algebra can touch almost every field of math, most notably algebra, geometry, differential equations, and statistics. Matrix methods are also fundamental to theoretical physics in the study of quantum mechanics. This course is a combination of three approaches: computational, theoretical, and applied. We begin with the mechanics of solving equations and matrix arithmetic, and progress into more abstract areas as we add structure to our knowledge. Applications will arise throughout the course in assignments and projects.
More on this wide area of study can be found in the linear algebra section of the Mathematical Atlas, and by following the many links in Eric Weisstein's World of Mathematics. Some of the history can be found via the above links or in the algebra section of the MacTutor History of Math Archive.
spring 2003 course description
spring 2003 syllabus
project guidelines
Mathematica files
spring 2003 class profile
course evaluation form
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