Linear equations were first introduced in high school algebra. Where we solved for several unknowns when we were given several equations. High school methods work fine for 1,2, or 3 unknowns, but the more unknowns and equations we have the longer it takes to find those unknowns. Linear algebra gives shortcuts for finding those unknowns. These techniques work if we have 2 unknowns or 203,456 unknowns, and if we run them on a computer, can have similar runtimes. The first two chapters of the textbook gives the fundamentals of what linear algebra is as well as the basic short cuts.
When solving a system of linear equations, the first step is to convert the equations into a matrix system: Ax=b where is the matrix of coefficients, x is a vector of the unknowns, and b is a vector of the constant terms.
Using the determinant of a system, one can see if the system is even solvable.
Linear Algebra is a part of Algebra. One of the main things that is studied in Algebra is Rings. A special type of Ring is a Vector Space.
In general a matrix applied to a vector or a matrix acting on a vector (ie: a matrix times a vector) changes the vector's magnitude (ie: length) and direction. An eigenvector is a vector that when the matrix is applied to it only has its magnitude multiplied by some factor. If that factor is positive the vector has only had its length changed, and if that factor is negative, the direction of the vector had been reversed as well as any changes to its length. That factor is called an eigenvalue. An eigenspace is the set of all eigenvectors that have the same eigenvalue.
If we have time, we'll get to this chapter. In this section, we'll look at some techniques (ie: QR, Least Squares, SVD) that are what people actually use. Typically the systems of equations (and matrices) are so large that using direct methods we've learned in previous section would be insane to use because the time it would take solve the systems would be astronomical. Here we'll look at alternatives that cut that runtime down.