MATH 323 Lecture 1

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Course Information

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

1. 7^th Week
2. 12^th Week
3. 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 $x_1,x_2,\dots ,x_n$ and functions $f_1(x_1,x_2,\dots ,x_n)=0,f_2(\dots ),\dots$.

• 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
  • $f(x_1,\dots ,x_n)=a_1{x}_1+a_2{x}_2+\dots +a_n{x}_n+b$
• 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 $a_1{x}_1+a_2{x}_2+\dots +a_n{x}_n=b$

$\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\dots +a_{1n}{x}_n=b_1\\ a_{21}{x}_1+a_{22}{x}_2+\dots +a_{2n}{x}_n=b_2\\ \dots \\ a_{m1}{x}_1+a_{m2}{x}_2+\dots +a_{mn}{x}_n=b_m\end{array}$

There are $m$ equations and $n$ unknowns. We will refer to this system as a $m\times n$ system.

Rules

1. The order in which any two equations are written may be interchanged
2. Both sides of an equation may be multiplied by the same non-zero real number
3. 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

1. $\{\begin{array}{l}{x}_1+2x_2=5\\ 2x_1+3x_2=8\end{array}$ (2 × 2, the unique solution to which is $x=(1,2)$)
2. $\{\begin{array}{l}{x}_1-x_2+x_3=2\\ 2x_1+x_2-x_3=4\end{array}$ (2 × 3, solutions are $\{(2,\beta ,\beta )\mid \mathrm{\forall }\beta \in \mathbb{ℝ}\}$
3. $\{\begin{array}{l}{x}_1+x_2=2\\ x_1-x_2=1\\ x_1=4\end{array}$ (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, $\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2=b_1\\ a_{21}{x}_1+a_{22}{x}_2=b_2\end{array}$

Plotting each equation yields a line.

1. If a unique solution exists, it will be at the coordinates of the lines' intersection ({x_1}^0, {x_2}^0
2. If empty solutions (inconsistent), both lines will be parallel
3. 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 ($m\times n$ and $k\times n$), they are called equivalent if they share the same solution set.

(Prof's notation: $I\sim II$)

Consider the following systems:

$\{\begin{array}{l}3x_1+2x_2-x_3=-2\\ x_2=3\\ 2x_3=4\end{array}$

$\{\begin{array}{l}3x_1+2x_2-x_3=-2\\ -3x_1-x_2+x_3=5\\ 3x_1+2x_2+x_3=2\end{array}$

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

Suppose we take two equations $f_i,f_j,0\le i,j\le m$ from a $m\times n$ system and replace $f_j$ by $f_j+\alpha f_i,\alpha \in \mathbb{ℝ}$, the original system and the new system are equivalent. The opposite direction also applies: $f_j=f_j-\alpha f_i$

Strict Triangular Form

Given a $m\times n$ system of equations where $m=n$, in $k$^th equation, the first $k-1$ variables/coefficient are all zero and the coefficient of $x_k$ is not 0

$\begin{array}{rl}{a}_{11}{x}_1+\dots +a_{1n}{x}_n& =b_1\\ 0x_1+a_{22}{x}_2+\dots +a_{2n}{x}_n& =b_2\\ 0x_1+0x_2+a_{33}{x}_3+\dots +a_{3n}{x}_n& =b_3\\ \dots \\ 0x_1+0x_2+\dots +0x_{n-1}+a_{mn}{x}_n& =b_{m-1}\end{array}$

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:

$\{\begin{array}{l}{x}_1+2x_2+x_3=3\\ 3x_1-x_2-3x_3=-1\\ 2x_1+3x_2+x_3=4\\ \end{array}$

$(2)-3(1)\leftarrow -7x_2-6x_3=-10$ $(3)-2(1)\leftarrow -x_1-x_3=-2$ $(3)-\frac{1}{7}(2)\leftarrow -\frac{1}{7}{x}_3=-\frac{4}{7}$

Solution is (3,-2,4)

Homework

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

Section 1.2: 3, 5egi, 8, 9