All courses › Linear Algebra and Differential Equations › Linear systems and rank

Linear systems and rank

The rank of a matrix is the number of pivot entries in its row echelon form, and tells us how many equations are actually independent. If a system has more unknowns than its rank, some variables become free.

rank(A)=number of pivots\text{rank}(A) = \text{number of pivots}the rank of A
free variables=n−rank(A)\text{free variables} = n - \text{rank}(A)nn is the number of unknowns

Symbols

nnnumber of unknowns

Example

A system with 3 equations and 5 unknowns has rank at most 3, so at least 5−3=25-3=2 variables are free.

If det⁡A=0\det A=0 for a square system, Ax⃗=0⃗A\vec x=\vec 0 also has non-trivial solutions.
Practise linear systems and eigenvalues for free →

← Eigenvalues and eigenvectors · Basis of a vector space →

Part of Linear Algebra and Differential Equations: Linear systems and eigenvalues.