# 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

(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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