Understand the equivalence between a system of linear equations, an augmented matrix, a vector equation, and a matrix equation.
Characterize the vectors such that is consistent, in terms of the span of the columns of
Characterize matrices such that is consistent for all vectors
Recipe: multiply a vector by a matrix (two ways).
Picture: the set of all vectors such that is consistent.
Vocabulary word:matrix equation.
Subsection2.3.1The Matrix Equation
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.
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.
Properties of the Matrix-Vector Product
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
Given a specific choice of what are all of the solutions to
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
We now have four equivalent ways of writing (and thinking about) a system of linear equations:
As a system of equations:
As an augmented matrix:
As a vector equation ():
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
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.
Recipe: The row-column rule for matrix-vector multiplication
The equivalence of 1 and 2 is established by this note as applied to every in
Now we show that 1 and 3 are equivalent. (Since we know 1 and 2 are equivalent, this implies 2 and 3 are equivalent as well.) If has a pivot in every row, then its reduced row echelon form looks like this:
and therefore reduces to this:
There is no that makes it inconsistent, so there is always a solution. Conversely, if does not have a pivot in each row, then its reduced row echelon form looks like this:
which can give rise to an inconsistent system after augmenting with
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”.
We will see in this corollary in Section 2.7 that the dimension of the span of the columns is equal to the number of pivots of That is, the columns of span a line if has one pivot, they span a plane if has two pivots, etc. The whole space has dimension so this generalizes the fact that the columns of span when has pivots.