Introduction

• for contact: Discord is also an option
• online HW: 5%; due every Monday
  • WebWork w/ some attempt
  • computing value
• offline PSETs; due every Wednesday
  • choose 4 problem; exam-like course
  • no extra harsh grading
  • 5% bonus for up to 4 bonus problems
• assignment submission: talk to TA
• A-range: w/ cutoff that's based on exam difficulty
  • ~= absolute grading in 1013/4

Notation:

• $\mathbb{R}$ denotes the real numbers.
• $\mathbb{Q}$ denotes the rational numbers p/q.
• $\mathbb{Z}$ denotes the integers $\{\dots ,-1,0,1,\dots \}$
• $\mathbb{N}$ denotes the nonnegative integers $\{0,1,2,\dots \}$.
• Ellipsis ($\dots$) notation: we write $a_1,a_2,\dots ,a_7$ instead of the full list .

Linear equation

• let $n\in {\mathbb{Z}}^{+}$; $x_1,x_2,\dots ,x_n$ be var. and $a_1,a_2,\dots a_n,b$ constants $\in \mathbb{R}$
  • 👨‍🏫 prefers $x_1,x_2,x_3$ than $x,y,z$
• linear equation in var $x_1,x_2,\dots x_2$ can be written in:
  • ${\sum }_{i=1}^n{a}_i{x}_i=b$
  • examples
    • $4x_1+5x_2-\pi x_3=-1$
    • $0=0$
    • $0=1$ // equation, but a false statement
      • corresponds to case $a_1=0,a_2=0,a_2=0,b=1$
  • 👨‍🎓as long as there is no exponent $\notin \{0,1\}$
    • and no var. is multiplied with another var.
    • or: var is not in exponent
    • $\sqrt{x^2}=>|x|$; not considered as linear
• linear system: a list of linear equations
• solution to linear system: assignment of val to var s.t. all equations in the system are simultaneously true
  • solution: non-zero if at least one assignment of var. $\ne 0$
  • e.g. $\{\begin{array}{l}{x}_1+x_2=7\\ x_2-x_1=1\end{array}$
    • has solution $(x_1,x_2)=(3,4)$
    • linear systems: equivalent if their solution are the same (or: have the same set of solutions)
      • and with the same variable sets
• any linear system: w/ 0, 1, or infinitely many solutions
  • if a system has 2 or more solutions -> has $\infty$ solutions
  • lin. sys w/ no solution: inconsistent
  • lin. sys w/ exactly 1 solution: consistent

Proof on No. of Solutions on Lin Alg Systems of Equation

• theorem: A linear system in two variables $x_1$ and $x_2$ has either 0, 1, or infinitely many solutions
• by geometry
  • a solution to one lin. equation: represents a point io a line
  • solution to a system of 2-var lin. eq.: represents intersection of all lines defined by the equations
  • however: collections of lines must either:
    • intersect at one point
    • intersect at no point
    • infinitely many solutions
      • i.e. all equations: represent the same line
  • 👨‍🎓 what if we are on spherical, or non-euclidean plane?
• by algebra
  • just check if lin. sys. has 2 different solution
  • given lin. sys., define the associated homogeneous system
    • where you set $b=0$
  • if $(s_1,s_2)$ is solution to the starting system
    • and $(t_1,t_2)$ solution to associated homogeneous system.
    • then $(s_1+t_1,s_2+t_2)$ is also the solution to starting system
    • as $\sum a_i(s_i+t_i)=\sum a_i{s}_i+\sum a_i{t}_i=b+0=b$
  • if homogenous system has a non-zero solution, then it has infinite many solutions
    • can be obtained by multiplying $t_i$ to any real number, and adding to original solution
  • if starting sys. has two different solutions: $(s_1,s_2)$ and $(r_1,r_2)$
    • then $(t_1,t_2,)=(s_1-r_1,s_2-r_2)$ is a non-zero solution to homogenous system
  • as $\sum a_i(s_i-r_i)=\sum a_i{s}_i-\sum a_i{r}_i=b-b=0$

Matrix

• matrix: rectangular array of numbers
  • e.g. $\left[\begin{array}{c}\sqrt{2}\end{array}\right]$, $\left[\begin{array}{cc}1.1& -1.1\\ 1.1& 1.2\end{array}\right]$, $\left[\begin{array}{ccc}1& 7& -1\\ 0& 4& 3\end{array}\right]$
• general matrix: denoted by $A=\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\end{array}\right]$
  • where $A_{23}$ is read as "A, two, three"
• matrix $A$ w/ $m$ rows & $n$ columns: called "$m\times n$" or "$m$-by-$n$"
  • index: starts from 0; top->down and left->right
  • entry in row $i$, col $j$ of matrix $A$: $A_{ij}$
    • or "entry of A in position $(i,j)$"
  • 👨‍🏫 position itself: also called "position $(i,j)$"
• zero matrix: matrix filled w/ 0

Augmented Matrix of the System

• essential information of sys. of equation: set of $a$s and $b$ in a matrix

  $$\begin{array}{rl}{x}_1-2x_2+x_3& =0\\ 2x_2+8x_3& =8\\ 5x_1-5x_3& =10\end{array}$$

• changes into $\left[\begin{array}{cccc}1& -2& 1& 0\\ 0& 2& -8& 8\\ 5& 0& -5& 10\end{array}\right]$ OR $\left[\begin{array}{cccc}1& -2& 1& 0\\ 0& 2& -8& 8\\ 5& 0& -5& 10\end{array}\right]$

• bar: can be anywhere, but often used to separate constant terms

• coefficient matrix of system: w/o constant terms

  $\left[\begin{array}{ccc}1& -2& 1\\ 0& 2& -8\\ 5& 0& -5\end{array}\right]$

Solving linear systems

• solve by transforming augmented matrix for certain operations

  1. add $-5\times$ equation 1 to equation 3

     $\left[\begin{array}{cccc}1& -2& 1& 0\\ 0& 2& -8& 8\\ 0& 10& -10& 10\end{array}\right]$

  2. multiply equation 2 by $1/2$

     $\left[\begin{array}{cccc}1& -2& 1& 0\\ 0& 1& -4& 4\\ 0& 10& -10& 10\end{array}\right]$

  3. add $-10\times$ equation 2 to equation 3

     $\left[\begin{array}{cccc}1& -2& 1& 0\\ 0& 1& -4& 4\\ 0& 0& 30& -30\end{array}\right]$

  4. multiple equation 3 by $1/30$

     $\left[\begin{array}{cccc}1& -2& 1& 0\\ 0& 1& -4& 4\\ 0& 0& 1& -1\end{array}\right]$

• augmented matrix of the system is triangular if all entries in position $(i,j)$ w/ $i>j$ are zero
• triangular system: can be solved easily from bottom-up
  • last equation: $x_3=-1$
  • which substitutes second eq. into $x_2-4x_4=4\Rightarrow x_2-4(-1)=4\Rightarrow x_2=0$
  • then, the first: $x_1-2x_2+x_3=x_1=2(0)+(-1)=0\Rightarrow x_1=1$

Row operations

• we have used following elementary row operations on the augmented matrix of the system

1. Replacement: replace one row by the sum of itself & a multiple of another row

   $\left[\begin{array}{cc}1& 2\\ 5& 6\end{array}\right]\to \left[\begin{array}{cc}1& 2\\ 6& 8\end{array}\right]$ or $\left[\begin{array}{cc}1& 2\\ 5& 6\end{array}\right]\to \left[\begin{array}{cc}501& 602\\ 4& 6\end{array}\right]$

2. Scaling: multiply all entries in a row w/ non-zero number

   $\left[\begin{array}{cc}1& 2\\ 5& 6\end{array}\right]\to \left[\begin{array}{cc}1& 2\\ -500& -600\end{array}\right]$

3. Interchange: swap two rows

   $\left[\begin{array}{cc}1& 2\\ 5& 6\end{array}\right]\to \left[\begin{array}{cc}5& 6\\ 1& 2\end{array}\right]$

Row equivalence

• two matrices: row equivalent if one can be transformed to the other
  • through a seq. of row operations
• each row operations are revertible
  • 👨‍🏫 can you show why?
  • replacement: as another row, used for addition, remains uncontaminated, you can subtract it back (with coefficient)
  • scaling: as one cal multiply reciprocal
  • interchange: as one can swap back