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.
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.
Let be an matrix, let be vectors in and let be a scalar. Then:
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
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.
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
We now have four equivalent ways of writing (and thinking about) a system of linear equations:
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.
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.
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.
If is an matrix with rows and is a vector in then
Let be a matrix with columns
Then
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 ).
Building on this note, we have the following criterion for when is consistent for every choice of
Let be an (non-augmented) matrix. The following are equivalent:
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 ”.
