Lecture 003

Gaussian Process

Gaussian Elimination: a systematic way of solving the system

not always possible, when impossible, get Row Echelon Form
always reaches echolon form
If the number of equations equals the number of variables, the system in non-singular

Pivot: leading coefficients

(Row) Echelon Form (ref):

Rows at the form (0, ..., 0 | x) are at the bottom if exists
any pivot cutting is in a column to the left from any pivot below it // QUESTION: WHAT? Explain? pivot of a nonzero row is always strictly to the right of the leading coefficient of the row above it.
If the system is non-singular: Echolon Form is Triangular
- there is exactly one solution
- every column has a pivot entry
If the system is singular: at least one (0, ..., 0 | x) (but the opposite is not true)
- if x == 0: infinite many solution if it is in $n$ by $n$ system
- if x != 0: no solution in any system

Reduced Row Echelon Form (rref): if furthermore all of the leading coefficients are equal to 1

Canonical Form: can be viewed as solution

Triangular System: special Echelon Form which indicate exactly one solution

$\begin{pmatrix} \begin{array}{ccccc|c} \circ & x & x & x & x & x\\ 0 & \circ & x & x & x & x\\ 0 & 0 & \circ & x & x & x\\ 0 & 0 & 0 & \circ & x & x\\ 0 & 0 & 0 & 0 & \circ & x\\ \end{array} \end{pmatrix}$

where $\circ$ represent non-zero, and $x$ can be zero or non-zero

Solving a system with infinite many solutions

If a system has infinite many solution, pick the variable that is not in Row-reduced form, then subsitute that with variable constant $P$ , solve for $P$ .

A by B matrix

a number of equation
b number of variables

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