I didn't go to the classes on this section, and now trying to study the book, I am completely lost. All i know is it has something to do with algebraic inequalities.
What is linear programming using the simplex method, according to linear algebra?
This is not something that can be explained here in a few lines. I suggest you start here http://en.wikipedia.org/wiki/Simplex_met... look up some of the references in the article. Maybe their approach will be enough different from your text to help.
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