We encounter yet another representation for a system of linear equations - will it never end?! This is the last we'll examine, and probably the most important. Theorem four pulls all these forms together: spans, pivots, linear combinations, and matrix equations collide!
``A fundamental idea in linear algebra is to view a linear combination of vectors as the product of a matrix and a vector.'' p. 40
Matrix/vector multiplication is defined. One form that I find particularly
useful is the so-called ``row-vector rule'': a row of the matrix slams into the
variable vector 
, to produce a single entry in the 
vector.
Definition: product of matrix A and vector
If A is an m x n matrix, with columns 
, 
,
, and if is in 
, then the product of A and
is the linear combination of the columns of A using the
corresponding entries in as weights; that is,
Example: #4, p. 47
We now have four ways of writing a system of equations(!), as given in
Theorem Three (p. 42): If A is an m x n matrix, with columns
, , , and if is in
, the matrix equation
has the same solution set as the vector equation
which, in turn, has the same solution set as the system of linear equations whose augmented matrix is
Example: #9, p. 47
Existence of solutions is given by the following theorem:
Theorem Four (p. 43): Let A be an m x n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false.
in 
, the equation has a
solution.
in is a linear combination of the columns of A.
.
Example: #14, p. 48
A handy way to think about matrix multiplication: Row-Vector rule for computing 
If the product is defined, then the ith entry in the vector
(yes, it's a vector!) is the sum of the products of corresponding
entries from row i of A and from the vector .
Example: Revisit #4, p. 47
Theorem Five (p. 45):
If A is an m x n matrix, 
and 
are vectors in
, and c is a scalar, then:
Example: #35, p. 49