We saw in the above example that the system of equations (3.2.2) is consistent. Equivalently, this means that the vector equation (3.2.1) has a solution. Therefore, the figure above is a picture of a consistent system of equations. Compare this figure below.

In order to actually solve the vector equation

one has to solve the system of linear equations

This means forming the augmented matrix

and row reducing. Note that the columns of the augmented matrix are the vectors from the original vector equation, so it is not actually necessary to write the system of equations: one can go directly from the vector equation to the augmented matrix by “smooshing the vectors together”.

Recipe: Solving a vector equation

In general, the vector equation

where are vectors in and are unknown scalars, has the same solution set as the linear system with augmented matrix

whose columns are the ’s and the ’s.

Now we have (at least) two equivalent ways of thinking about systems of equations:

Augmented matrices.

Linear combinations of vectors (vector equations).

The second is more geometric in nature: it lends itself to drawing pictures.

Subsection3.2.2Spans

It will be important to know what are all linear combinations of a set of vectors in In other words, we would like to understand the set of all vectors in such that the vector equation (in the unknowns )

has a solution (i.e. is consistent).

Definition

Let be vectors in The span of is the collection of all linear combinations of and is denoted In symbols:

We also say that is the subset spanned by or generated by the vectors

The above definition is the first of several essential definitions that we will see in this textbook. They are essential in that they form the essence of the subject of linear algebra: learning linear algebra means (in part) learning these definitions. All of the definitions are important, but it is essential that you learn and understand the definitions marked as such.

Set Builder Notation

The notation

reads as: “the set of all things of the form such that are in ” The vertical line is “such that”; everything to the left of it is “the set of all things of this form”, and everything to the right is the condition that those things must satisfy to be in the set. Specifying a set in this way is called set builder notation.

All mathematical notation is only shorthand: any sequence of symbols must translate into a usual sentence.

Three characterizations of consistency

Now we have three equivalent ways of making the same statement:

A vector is in the span of

The vector equation
has a solution.

The linear system with augmented matrix
is consistent.

Equivalent means that, for any given list of vectors either all three statements are true, or all three statements are false.

Pictures of Spans

Drawing a picture of is the same as drawing a picture of all linear combinations of