What is Linear Algebra?

You are familiar with the equation of a line you learned when you first took Algebra: y=mx+b, where all (x,y) are the points on the line themselves, m is the slope, and b is the y-intercept. Geometrically, this equation does represent a straight line, but unless b=0, f(x)=mx+b is not a linear function, instead we call it an affine function.

When you think about polynomials of x, remember that we call x0 the constant term, x the linear term, x2 the quadratic term, etc. A linear function, or a linear map, as it is more commonly called, is a function that consists of the linear term only, and nothing else. We care about linear maps because they are the only functions of real numbers where for any a, b, and c, f(a+b)=f(a)+f(b) and f(c*a)=c*f(a).

That sounds boring, right? The additive and scalar multiplication properties of the function itself are nice, but all the function f(x)=mx does is take a number and multiply it by m, so there's not much to discuss. However, we've only been looking at functions of single variables, and in Linear Algebra we are much more interested in linear maps of vectors. In physics, a vector is a quantity that has magnitude and direction, but in math, a vector is simply a list of numbers. In a three-dimensional real vector space, (1,2,3) is one example of a vector.

You will spend the first few weeks in Linear Algebra talking about vectors and vector spaces (sets of vectors that satisfy certain properties), but then you will start talking about linear maps themselves. A linear map of vectors is commonly represented as an array of numbers called a matrix. This matrix, for example:

takes three-dimensional vectors and maps them to two-dimensional vectors.

You probably recognize matrices from using Cramer's Rule or reduced row echelon form to solve a system of equations in high school. These methods involve using square matrices (linear maps that map a vector space to itself), and they are interesting and useful enough to get a special term: linear operators or simply operators. There are several reasons that operators are more useful, for example it is possible for an operator to be inverted, and you will learn more about them when you take the class.