Systems of linear equations

Overview

flowchart TD
    A[Linear Equation System] --> B(Solving for unknown <br/>variables)
    B --> |Echelon Form| C[3 different types of EF<br/> consistent vs. inconsistent?]
    C -->|consistent if| D[Rank Criterion <br/> rkA = rk A,b <br/> for Ax=b]
    D --> K[Rank = #Pivots <br/> Nullity = #FREEparameter]
    A --> E[Inhomogeneous Linear System <br/> Ax=b]
    E --> F[Solution Set: Particular + <br/> Homogenous System]
    A --> G[Matrix Form <br/> Vector Form]
    A --> H[Square System]
    H --> |check with <br/> determinants| HA[invertible,regular <br/> non-singular <br/> if AX=I]
    HA --> |if no inverse|HB[singular <br/> non-invertible]
    HA --> |double-augmented <br/> matirx|HC[calculate with LSE]
    HA --> |Corollary|HE[if A and B invertible <br/> AB invertible]
    HC --> HD[if no pivot]
    HD --> HB
    H --> I[Unique Solution. <br/> Hom.S. only trivial sol]
    I --> J[except if any zero rows]
    A --> L[vectors linear independent?]
    A --> M[Linear transformation <br/> with non-standard basis <br/> and mapping rule L]
    M --> |to find:|MA[Coordinates of <br/>images vector L: v_j]
    MA --> |each image basis <br/>has own LSE|MAA[find linear combination of <br/> basis that form <br/> image of basis. <br/> Coefficients = coordinates]

Figure 1: A flow chart displaying for what to use linear system equations and gaussian elimination for solving the system. Only lecture relevant use cases are mentioned.

Connection to linear maps

• Linear systems of equations can be described by matrices
• Matrices can be interpreted as linear maps

Three Types of Echelon Form (EF)

1. Pivot element in every row and column.
   • Squared form only possible when $r=c$
   • Last pivot is isolated and can be substituted
   • unique solution
2. Pivot in every row but not in every column.
   • more unknown than equations: $r<c$
   • free parameter (the column without pivot)
   • infinitive solutions
3. Rows without pivot.
   • $0=b$
     • if $b \not= 0$ no solution since not consistent
     • if $b = 0$ no information. We can end up with form 1 or 2.

General Solution

Typically you have some pivots missing (in EF) and some rows with 0 = 0.

You express the pivots columns with the missing pivot column (here z).

• First we use substitution to formulate the solution for the pivot columns (x and y):

• Now lets determine the solution set:

The free parameters are used to describe the kernal of a matrix using homogenous solution.

Applied Problems: Process optimization

Many problem can be formulated as system of linear equations:

Form linear system of equations

Looking at the equation we make following observations:

• we want to find $x_A,x_B,x_C$ that fits the equations as a solution.
• we add the terms because the individuel maschines can’t process the chemicals in parallel so we have to add the time
• the coefficients of the unknown show a relationship of time per ton
• $\frac{time}{tone}*time = time$
• we have three unknown but only 2 equations: 1 free parameter -> infinitive solutions
• $x_A,x_B,x_C$ can only be positive since you can’t produce negative tones.

Solving the system

Solving the linear system will end up with a general solution:

and since the unknown can’t be negative we have to see for which range of $x_C$ $x_A$ and $x_B$ are not negative:

we can chose any tonne of $x_C$ between 0 and 15.

Determinats

For 2x2 squared matrices:

ad - bc

For 3x3 squared matrices:

Rule of Sarrus

Other n x n cases

bring to upper triangular matrix form by row operations and switching:

• for each switch sign of det changes!
• if one column with no pivot element: $det(0)$
• if upper triangular: determinant is product of main diagonal elements

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