Handouts for Class & Supplemental Work

Chapter 1 - Linear Equations
Chapter 2 - Matrix Algebra
Chapter 3 - Determinants
Chapter 4 - Vector Spaces
Chapter 5 - Eigenvalues & Eigenvectors
Chapter 6 - Orthogonality

Chapter 1: Linear Equations

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.

Details about onto, one-to-one, and square matrices
Determining onto or one-to-one
Lecture Notes
Section 1.1
Section 1.2
Section 1.3 & 1.4
Section 1.5
Section 1.6
Section 1.10

Chapter 2: Matrix Algebra

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.

Entering matrices into a calculator
Lecture Notes
Section 2.6 & 2.7

Chapter 3: Determinants

Using the determinant of a system, one can see if the system is even solvable.

Chapter 4: Vector Spaces

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.

Chapter 5: Eigenvalues & Eigenvectors

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.

Lecture Notes
Section 5.1
Section 5.2 and an application
Section 5.3
Section 5.4