1. Basics

Lineare Gleichungssysteme & Homogene Systeme

Lineare Gleichungssysteme

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Ax = b
A ∈ ℝ^m×n, x ∈ ℝⁿ, b ∈ ℝ^m

A linear system of m equations in n unknowns can be written as a matrix equation Ax = b. The augmented matrix [A|b] encodes the full system. We solve it by bringing it to row echelon form (Zeilenstufenform) via Gaussian elimination.

What is Rang (Rank)?

The rank of a matrix is the number of pivot positions (leading non-zero entries) in row echelon form. It tells you the number of independent equations — equations that actually constrain the solution. If some rows become all zeros, those equations were redundant.

Preset examples:

Custom [A|b] matrix:

Size:

Lösungsfälle (Solution Types)

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Keine Lösung: r < m und b_i ≠ 0 für ∃i > r
Eindeutige Lösung: r = n = m
Unendlich viele: r < n und b_i = 0 ∀i > r

After Gaussian elimination, the rank r determines the solution type. If a zero row has a non-zero entry in b, the system is inconsistent (no solution). If r = n (as many pivots as unknowns), the solution is unique. If r < n, there are n − r free parameters — infinitely many solutions.

Solution type:

How to count free parameters

After row echelon form: free parameters = n − r (unknowns minus rank). Columns WITHOUT a pivot correspond to free variables that can take any value. The other variables (pivot variables) are determined by back-substitution.

Rang, Det & Invertierbarkeit

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Für A ∈ ℝ^n×n: rang(A) = n ⇔ det(A) ≠ 0 ⇔ A invertierbar

For a square matrix A ∈ ℝ^n×n, the following are ALL equivalent — if any one is true, all are true. If any one fails, they all fail.

Matrix to test:

rang(A) = n (voller Rang)
det(A) ≠ 0
A ist invertierbar
Ax = b ist für jedes b lösbar
Ax = b hat genau eine Lösung
ker(A) = {0} (nur triviale Lösung)
Spalten sind linear unabhängig
Spalten bilden eine Basis von ℝⁿ
Kein Eigenwert ist 0

Why are these all equivalent?

They all describe the same thing from different angles: the columns of A span all of ℝⁿ. If they do, you can reach any b (Ax=b solvable), the map is injective (unique solution), there's nothing in the kernel, the determinant (which measures volume scaling) is non-zero, and no eigenvalue is 0 (no direction gets crushed to zero).

Homogene Systeme (Ax = 0)

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Ax = 0
Triviale Lösung x = 0 existiert immer

A homogeneous system Ax = 0 always has at least one solution: x = 0 (the trivial solution). The question is whether there are non-trivial solutions (x ≠ 0).

rang(A) = n ⇒ nur x = 0
rang(A) < n ⇒ ∞ viele Lösungen, dim(ker) = n − r

Example:

Der Kern (Kernel / Nullspace)

The kernel ker(A) = {x ∈ ℝⁿ | Ax = 0} is a subspace of ℝⁿ. Its dimension is n − r (number of free parameters).

Geometrically: the kernel is the set of all vectors that A "crushes to zero." If A has full rank, only the zero vector gets crushed. If rank is deficient, an entire line, plane, or higher-dimensional subspace gets mapped to the origin.

Connection to general Ax = b

If x_p is any particular solution of Ax = b, then the general solution is:

x = x_p + ker(A)

Every solution of Ax = b equals a particular solution plus something from the kernel. This is why: unique solution ⇔ ker(A) = {0}.

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Chapter 1: Basics — Lineare Gleichungssysteme

Section 1 | Select an example