Math 482 (Lecture 4): The simplex method I: linear
algebra reminders and basic (feasible) solutions
Today: We're going to start discussing the elements needed in the simplex method.
Reminders about linear algebra
:
Linear algebra is concerned about solving systems of linear equations
Ax =b
where A is an mxn matrix in n unknowns.
* The basic tool is
Gaussian elimination
as encoded by
augmented matrices
.
* The
rank
of the matrix A is the number of linearly independent columns it has. This is
at most m.
Assumption:
A actually has rank m. (Otherwise there are some redundant equations we
can remove/or inconsistent equations.)
* Now I'm going to start introducing terminology that's specific to describing the simplex
method.
Definition:
A
basis
of A is a collection of m linearly independent columns
1≤ j
1
< j
2
≤... ≤ j
m
of our matrix A.
In class exercise:
Consider the matrix
1
1
1
0
1
0
What are the bases? What are the nonbases?
If an mxn matrix has at least one basis, can you always solve Ax=b?
For example, can you solve
1
1
0
1
0
1
1
0
1
Brief solution:
Columns
{1,4,5}, {1,2,4}, {1,2,5}, {2,3,4}, {2,3,5}, {2,4,5}
are bases. Non bases are all other choices of columns, e.g., {1,2,3}.
You can determine this by either row reduction or taking a determinant.
When you have a basis, you can always solve Ax=b. One way to say this is that the span
of the basis is all of R
m
.