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Basis & Dimension

Basis and dimension calculator for vectors. Enter several vectors to test independence and compute a basis and the dimension of the column space—great for linear algebra practice.

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dim V = rank(vector set)
semicolon separates vectors

📖 Tutorial｜Basis & Dimension

1. Definition
A maximal linearly independent set in V is a basis; its size is the dimension dim V.
2. Symbols
SymbolMeaning
basismaximal independent set
dim Vdimension = rank
spanlinear combinations fill the space
3. How this tool works
  • A basis is independent and spans V;
  • Dimension equals the rank of the set;
  • Bases are not unique, but dimension is.
4. Steps
  1. Enter one vector per row;
  2. Click Calculate for the dimension and a basis.
5. Example
Example: v₁=(1,0),v₂=(0,1),v₃=(1,1). v₃=v₁+v₂, rank=2, dim=2; take v₁,v₂ as a basis.
6. Pitfalls
Bases are not unique, but dimension is;
Dimension never exceeds the vector dimension;
This tool uses the first r independent vectors as a sample basis.

❓ FAQ｜Basis & Dimension

What is a basis?
A maximal linearly independent set in the vector space.
How to compute dimension?
It equals the rank of the set.
Is the basis unique?
No, but its size (dimension) is.
Standard basis of Rⁿ?
e₁,…,eₙ, with dim=n.
Dimension of the null space?
n − rank(A).