Linear Algebra, An Introduction

Introduction

A vector space is a left or right unitary module over a division ring or a field.

What??

Ok, so you don't know anything about rings, fields, or modules.  Let's describe it in another way.

A vector is an arrow that starts at the origin and extends to a certain point in space.  The coordinates of the end point define the vector.  If you're working in the plane, each vector consists of two numbers.  The vector [1,1] points into the first quadrant, while the vector [-1,1] points into the second.  Vectors in 3 space consist of 3 numbers, and so on.

The great thing about vectors is, they can be scaled and added together, consistent with the definition of a module.  In our geometric example, multiplication by a constant k is done per coordinate, and vectors are added together by adding coordinates.  Thus [1,1] + [-1,1] = [0,2], a longer vector that points straight up the y axis.

The set of all vectors, closed under addition and scaling, is called a vector space.  If we start with [1,1] and [-1,1] in the plane, and use real coefficients, the entire plane is the vector space.  Verify that every (x,y) is a linear combination of these two vectors.  Yet this is merely one example.  The coefficients could be rational, whence the vector space consists of points in the plane with rational coordinates.  Or the coefficients could be complex numbers, giving a vector space that has two dimensions relative to complex numbers, and 4 "real" dimensions.  Still other vector spaces have infinitely many dimensions, where "infinite" could be any cardinality.  The only requirement is that the coefficients used to scale the vectors have inverses.  If you can double a vector, you can cut it in half, and so on.

A subspace is a vector space contained within another vector space.  The same scaling coefficients are assumed.  The rationals along the x axis are a subset of the entire xy plane, but not a subspace, since the plane uses real numbers to scale its vectors.  You need the entire "real" x axis; that's a subspace.  In other words, the subspace is a submodule.