##### Objectives
1. Understand the equivalence between a system of linear equations, an augmented matrix, a vector equation, and a matrix equation.
2. Characterize the vectors such that is consistent, in terms of the span of the columns of
3. Characterize matrices such that is consistent for all vectors
4. Recipe: multiply a vector by a matrix (two ways).
5. Picture: the set of all vectors such that is consistent.
6. Vocabulary word: matrix equation.

# Subsection2.3.1The Matrix Equation Ax = b .

In this section we introduce a very concise way of writing a system of linear equations: Here is a matrix and are vectors (generally of different sizes), so first we must explain how to multiply a matrix by a vector.

When we say is an matrix,” we mean that has rows and columns.

##### Definition

Let be an matrix with columns

The product of with a vector in is the linear combination

This is a vector in

In order for to make sense, the number of entries of has to be the same as the number of columns of we are using the entries of as the coefficients of the columns of in a linear combination. The resulting vector has the same number of entries as the number of rows of since each column of has that number of entries.

If is an matrix ( rows, columns), then makes sense when has entries. The product has entries.

##### Definition

A matrix equation is an equation of the form where is an matrix, is a vector in and is a vector whose coefficients are unknown.

In this book we will study two complementary questions about a matrix equation

1. Given a specific choice of what are all of the solutions to
2. What are all of the choices of so that is consistent?

The first question is more like the questions you might be used to from your earlier courses in algebra; you have a lot of practice solving equations like for The second question is perhaps a new concept for you. The rank theorem in Section 2.9, which is the culmination of this chapter, tells us that the two questions are intimately related.

##### Matrix Equations and Vector Equations

Let and be vectors in Consider the vector equation

This is equivalent to the matrix equation where

Conversely, if is any matrix, then is equivalent to the vector equation

where are the columns of and are the entries of

##### Four Ways of Writing a Linear System

We now have four equivalent ways of writing (and thinking about) a system of linear equations:

1. As a system of equations:
2. As an augmented matrix:
3. As a vector equation ():
4. As a matrix equation ():

In particular, all four have the same solution set.

We will move back and forth freely between the four ways of writing a linear system, over and over again, for the rest of the book.

##### Another Way to Compute Ax

The above definition is a useful way of defining the product of a matrix with a vector when it comes to understanding the relationship between matrix equations and vector equations. Here we give a definition that is better-adapted to computations by hand.

##### Definition

A row vector is a matrix with one row. The product of a row vector of length and a (column) vector of length is

This is a scalar.

##### Recipe: The row-column rule for matrix-vector multiplication

If is an matrix with rows and is a vector in then

# Subsection2.3.2Spans and Consistency

Let be a matrix with columns

Then

##### Spans and Consistency

The matrix equation has a solution if and only if is in the span of the columns of

This gives an equivalence between an algebraic statement ( is consistent), and a geometric statement ( is in the span of the columns of ).

##### When Solutions Always Exist

Building on this note, we have the following criterion for when is consistent for every choice of

##### Proof

Recall that equivalent means that, for any given matrix either all of the conditions of the above theorem are true, or they are all false.

Be careful when reading the statement of the above theorem. The first two conditions look very much like this note, but they are logically quite different because of the quantifier “for all ”.