# MATH 323 Lecture 1

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## Contents

## Course Information

2 in-class exams and a final (all in C E 222)

- 7
^{th}Week - 12
^{th}Week - TBA

Basic Scientific Calculator is allowed on exam (TI-34)

## Chapter 1: Matrices and Systems of Equations

A **system** is a set of unknown variables and functions .

**Linear functions**are function for which the exponent on all unknown variables (x's) is 1. Can be plotted as a line/plane in 2D/3D space**Quadratic functons**are non-linear functions for which the largest exponent on all x's is 2

Consider **systems of linear equations**, each function is often written in form

There are equations and unknowns. We will refer to this system as a system.

### Rules

- The order in which any two equations are written may be interchanged
- Both sides of an equation may be multiplied by the same non-zero real number
- A multiple of one equation may be added to / subtracted from another equation

These are more applicable to lesser systems of equations, and leads to a system that is equivalent to the original.

### Examples

- (2 × 2, the unique solution to which is )
- (2 × 3, solutions are
- (3 × 2, no solution that satisfies answers)

In examples 1 & 2, the systems were **consistent**, but since the solution set for 3 was empty, it is **inconsistent**

### 2 × 2 Systems

In general,

Plotting each equation yields a line.

- If a unique solution exists, it will be at the coordinates of the lines' intersection ({x_1}^0, {x_2}^0
- If empty solutions (inconsistent), both lines will be parallel
- If infinite solutions, both lines are the same line

### Equivalent Systems

Given 2 systems I and II, with potentially different number of equations but same number of unknowns ( and ), they are called **equivalent** if they share the same solution set.

(Prof's notation: )

Consider the following systems:

The two systems are equivalent because they both have unique solutions (-2, 3, 2)

Suppose we take two equations from a system and replace by , the original system and the new system are equivalent. The opposite direction also applies:

### Strict Triangular Form

Given a system of equations where , in ^{th} equation, the first variables/coefficient are all zero and the coefficient of is not 0

There is always a **unique solution** to a system of this form.

Solve the bottom 1-term equation, then plug that back into the previous equation, and continue **back-substitution** until all equations are satisfied.

Goal of rules is to reduce a system into strict triangular form:

Solution is (3,-2,4)

## Homework

Section 1.1: 1c, 2, 3, 5cd, 6ceh, 7, 8

Section 1.2: 3, 5egi, 8, 9