Math 308: Matrix Algebra Chapter 1.2 Notes
Chapter 1.2
1.2 Notes (Class 3 & Textbook)

Two linear systems are said to be equivalent if they have the same set of solutions.
 Can transform a linear system cusing a sequence of elementary operations (All are reversable):
 Interchange two rows.
Geometrically changes the order of graph.  Multiply a row by a nonzero constant.
Geometrically changes nothing.  Add a multiple of one row to another.
Geometrically taking solution and rotating around point of solution.
 Example:
 Interchange two rows.

A matrix contains all the coefficients of a linear system, including the constant terms on the right side of each equation, it is called an augmented matrix

The forward phase is Gaussian elimination, transforming the matrix to echelon form, and the backward phase, which completes the transformation to reduced echelon form.

The system is inconsistent if it has following form:
Gaussian elimination & Gauss–Jordan elimination example:
 Gaussian elimination:
 Gauss–Jordan elimination
Reduced echelon form: leading variable need to be one
Class 4
Example 1: Gausselimination:
$
\begin{cases}
3x_13x_2+9x_3=24\\
2x_12x_2+7x_3=17\\
x_1+2x_24x_3=11
\end{cases}
$
Check: plug $x_1=3, x_2=2, x_3=1$ into left sides of orginal equations.
Example 2
$\begin{cases}
x_1+x_2=2\\
2x_1+3x_2=3\\
x_1+3x_2=0\\
3x_1+6x_2=4
\end{cases}
$
No solution: inconsistent.
Example 3: Hamogeneous
Hamogeneous: always consistent, all x = 0 is a solution.
$x_1$ and $x_2$ are local variables.
$x_3$ and $x_4$ are free variables.
Let $x_3 = S_1 $ and $x_4 = S_2$ as free parameters.
Solution:
Theorem: A system of equations either has no solutions, exactly one solution, or infinitely many solutions.
Given any system of equations, form the augmented matrix and perform GaussJordan elimination: getting unique equivalent matrix in reduced echelon form.
Case:
1. Row with zeros ending with constant c != 0  No solution (See Textbook Notes above)
0 0 ... 0  c
2. No such rows
a) There is no free variable, then there is a unique solution.
b) There is one or more free variables, then there are solution sets (infinitly many solutions).