# 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):
1. Interchange two rows.
Geometrically changes the order of graph.
2. Multiply a row by a nonzero constant.
Geometrically changes nothing.
3. Add a multiple of one row to another.
Geometrically taking solution and rotating around point of solution.
• Example:

• 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: Gauss-elimination:

$\begin{cases} 3x_1-3x_2+9x_3=24\\ 2x_1-2x_2+7x_3=17\\ -x_1+2x_2-4x_3=-11 \end{cases}$

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 Gauss-Jordan 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).