Abstract We define linear equations, both homogeneous and inhomogeneous, and describe what is certainly the oldest problem in linear algebra: finding the solutions of a system of linear equations.

Linear algebra’s main focus is the study of a special category of transformations: the linear transformations. Linear transformations include rotations, dilations (stretches), shears, and more.

Okay, we’ve not got most of the basics down that we’ll need to start solving systems of linear equations using linear algebra techniques so it’s time to move onto the next section.

Simply put, Linear Algebra deals with classifications of linear and/or quadratic equations, or systems of such equations. One might think that all that equations do is ask: Solve us! Unfortunately this attitude …

When a function of vectors obeys the additivity and homogeneity properties we say that it is linear (this is the \linear" of linear algebra). Together, additivity and homogeneity are called linearity.

In this course, we'll learn about three main topics: Linear Systems, Vector Spaces, and Linear Transformations. Along the way we'll learn about matrices and how to manipulate them.

Some linear combination of the 100 columns is the column of zeros. A very singular matrix has all ones: A = ones (100). A better example has 99