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Linear Independence

Linear-independence checker. Enter vectors (one per row) to test dependence and report the rank—for vector sets.

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rank = #vectors ⇒ independent
spaces for entries, semicolons for vectors

📖 Tutorial｜Linear Independence

1. Definition
If some k₁…kₙ not all zero satisfy k₁v₁+…+kₙvₙ=0, the set is linearly dependent; otherwise independent.
2. Symbols
SymbolMeaning
v₁…vₙvector set
rankrank of the matrix
nnumber of vectors
3. How this tool works
  • Arrange vectors as rows of a matrix;
  • rank = n ⇒ independent;
  • rank < n ⇒ dependent.
4. Steps
  1. Enter one vector per row;
  2. Click Calculate for the rank and verdict.
5. Example
Example: v₁=(1,2),v₂=(2,4),v₃=(3,6). v₂=2v₁, v₃=3v₁, rank=1<3, dependent.
6. Pitfalls
More vectors than dimensions means dependent;
A set containing the zero vector is dependent;
Rank is the key criterion.

❓ FAQ｜Linear Independence

What is linear dependence?
A nontrivial combination equals zero.
How to test?
Arrange as a matrix; rank = n means independent.
More vectors than dimensions?
Always dependent.
Set containing zero vector?
Always dependent.
Rank and independent set?
Rank equals the size of a maximal independent subset.