Edwards-Penney Chapter 3, sections 3.1, 3.2, 3.3
Topics, Definitions, Theorems

 3.1
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 Linear equations, 2x2 system
 Three possibilities: Unique, No sol, Infinitely many
 Method of elimination, by examples
 Def: Parameter means a new independent variable valid for real numbers
 Def: Elementary operations (swap, combo,mult)
 Linear 3x3 systems

 3.2
====
 Matrix form of a linear system Ax=b
 Def: Coefficient matrix A, elements, rows, columns
 Def: Column vector, augmented matrix
 Def: Elementary row operations (swap, combo, mult)
 Def: Row-equivalent matrices are two matrices obtained from an original
      matrix by using only Toolkit operations swap, combo, mult. 
 Theorem 1. Any two solution frames obtained by Toolkit operations
            (combo, swap, mult) have the same solution set. In short,
            the Toolkit does not create or destroy solutions.

 Def: Leading variable and free variable.
 Def: An Echelon matrix B represents a system of equations Bx=0, where x
      has components x[1], x[2], x[3] ...; each nonzero equation has a
      leading variable x[k], for some k, and all equations following it
      do not contain the variables x[1] to x[k]. The nonzero equations
      are listed first and the zero equations last.
     
 Def: Back-substitution in Edwards-Penney means:
    1. Applies only to an echelon matrix. 
    2. Identify leading and free variables.
    3. Replace all free variables by invented symbols (parameters).
    4. Use the variable list in reverse order. Start with the last
       equation and solve for the leading variable. Repeat for the next
       leading variable in reverse list order, back-substituting leading
       variables from all previously treated equations (last equation to
       present). 
    5. Display the answer in variable list order, with only constants
       and invented symbols on the right of the equal sign.
 ALGORITHM. Gaussian Elimination.
    The book's "algorithm" applied to Ax=b finds from the augmented
    matrix C=<A|b> an echelon matrix B. Then back-substitution as
    defined above finds the answer x. About all the book tries to do is
    describe the examples preceding the algorithm.

 3.3
====
 Def: Reduced row-echelon matrix (rref)
 ALGORITHM. Gauss-Jordan Elimination.   
    Summary: Given matrix C (augmented or not) transform to echelon form
             by swap, combo, mult. Divide each nonzero row by its
             leading entry to get a leading one in each nonzero row. A
             column containing a leading entry (i.e. a leading one) is
             called a pivot column. Use COMBO to make all other pivot
             column entries equal to zero.

 Theorem 1. The rref is unique.
 Theorem 2. The three possibilities.
 Theorem 3. If variable x has more entries than there are equations in 
            system Ax=0, then the system has infinitely many solutions.
 Theorem 4. Let A be square nxn. The equation Ax=0 has only the solution 
            x=0 if and only if rref(A) is the identity matrix.