# Vectors and spaces

Contents

Let's get our feet wet by thinking in terms of vectors and spaces.

We will begin our journey through linear algebra by defining and conceptualizing what a vector is (rather than starting with matrices and matrix operations like in a more basic algebra course) and defining some basic operations (like addition, subtraction and scalar multiplication).

Given a set of vectors, what other vectors can you create by adding and/or subtracting scalar multiples of those vectors. The set of vectors that you can create through these linear combinations of the original set is called the "span" of the set.

If no vector in a set can be created from a linear combination of the other vectors in the set, then we say that the set is linearly independent. Linearly independent sets are great because there aren't any extra, unnecessary vectors lying around in the set. :)

In this tutorial, we'll define what a "subspace" is --essentially a subset of vectors that has some special properties. We'll then think of a set of vectors that can most efficiently be use to construct a subspace which we will call a "basis".

In this tutorial, we define two ways to "multiply" vectors-- the dot product and the cross product. As we progress, we'll get an intuitive feel for their meaning, how they can used and how the two vector products relate to each other.

This tutorial is a bit of an excursion back to you Algebra II days when you first solved systems of equations (and possibly used matrices to do so). In this tutorial, we did a bit deeper than you may have then, with emphasis on valid row operations and getting a matrix into reduced row echelon form.

We will define matrix-vector multiplication and think about the set of vectors that satisfy Ax=0 for a given matrix A (this is the null space of A). We then proceed to think about the linear combinations of the columns of a matrix (column space). Both of these ideas help us think the possible solutions to the Matrix-vector equation Ax=b.