$Vector Basis$

Vector Basis

Check whether vectors form a basis and decompose a vector in the basis, with the solution.

What to find:

Check whether the vectors form a basis

Decompose a vector in the basis

Coordinates with spaces, vectors with semicolons: 1 0 1; 0 1 1; 1 1 0. Up to 5 vectors and coordinates, fractions as 1/2.

This calculator checks whether vectors form a basis and decomposes a vector in that basis.

Vectors form a basis if there are as many of them as coordinates and the determinant of their matrix is not zero. The decomposition coefficients come from solving a system of linear equations.

Example. The vectors (1, 0) and (1, 1) form a basis because the determinant is 1. The vector (3, 2) decomposes as (3, 2) = 1 · (1, 0) + 2 · (1, 1).