Network Flows

Networks are represented with notes connected by edges.

Everything that goes in goes out, think of water in pipes

${\text{Flow}}_{\text{in}}={\text{Flow}}_{\text{out}}$

Less rows than variables means an infinite number of solutions

Operations on Matrices

adding subbing and multiplying boxes

i can index a matrices like an array

Transpose

$A_{ij}^T$ is transposing the matrix to $A_{ji}$

Matrix column Multiplication

https://assets.tivadardanka.com/2022_matrix_multiplication_def_01_b93c4c0ea2.png

AxB != BxA

A(BC) = (AB)(C)

A(B+C) = AB + AC != BA + CA (B+C)A = BA + CA

Itendty Matrix

The nxn identity matricex I_N is the matricx satisfiying

$(I_n{)}_{ij}$ = {0 if i != j and 1 if i = j}

so its like a matrix thats all pivots

AI

$$\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$$

Elementary matrices

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