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Homogeneous Linear Systems

Homogeneous linear system solver. Enter A to classify the solution, find a nullspace basis and general solution—with rank notes—for linear systems.

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Ax = 0
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Homogeneous Linear Systems

1. Definition
A system with zero RHS, Ax = 0, is homogeneous. It always has the zero solution x=0.
2. Symbols
SymbolMeaning
rank(A)rank of coefficient matrix
nnumber of unknowns
nullityn−rank, size of basis
3. How this tool works
  • A homogeneous system always has the zero solution;
  • Nontrivial solutions exist iff rank(A) < n;
  • A basis of the solution space has n−rank(A) vectors.
4. Steps
  1. Enter A;
  2. Click Calculate to see rank, nullity and whether nontrivial solutions exist.
5. Example
Example: A=1 1 −1;2 2 −2. rank=1<3, so nontrivial solutions exist; basis has 2 vectors.
6. Pitfalls
A homogeneous system is never inconsistent;
n−rank is the number of free variables;
A basis is not unique, but its size is.

❓ FAQ｜Homogeneous Linear Systems

Does a homogeneous system always have a solution?
Yes, at least x=0.
When are there nontrivial solutions?
When rank(A) < n.
How many vectors in a basis?
n − rank(A).
What is nullity?
n − rank(A), the number of free variables.
When does a square homogeneous system have only zero?
When det A ≠ 0.