Linear Independence Calculator

Work out linear independence instantly with clear inputs, formula shown and shareable results.

About Linear Independence Calculator

Vectors are linearly independent when the only combination giving zero is the trivial one. That happens exactly when the rank equals the number of vectors, or equivalently when a square arrangement has non-zero determinant.

Formula

Independence test

independent ⇔ rank = number of vectors ⇔ det ≠ 0 for a square set

Worked example

With the default values below, Linear Independence Calculator returns Result of The vectors are linearly dependent. Change any input above and every figure here is recalculated in your browser.

Inputs used

Vectors listed one after another: 1, 0, 2, 0, 1, 3, 1, 1, 5
Entries per vectoraccepts 1–8: 3

Results

Result: The vectors are linearly dependent
Rank of the set: 2
Vectors given: 3
Determinant if square (0 otherwise): 0.000000

Step by step

1Values used
2Independence test
3Result= The vectors are linearly dependent
4Rank of the set= 2
5Vectors given= 3
6Determinant if square (0 otherwise)= 0.000000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For (1,0,2), (0,1,3) and (1,1,5)?

The third is the sum of the first two, so the rank is 2 and they are dependent.

Can more than n vectors be independent in n dimensions?

No. Any set of more than n vectors in n-dimensional space is necessarily dependent.