Summary

Rank-nullity Theorem

Definition

Axiom

For any $n\times m$ matrix $A$, the rank-nullity theorem states that the sum of the rank of $A$ and the dimension of the null space of $A$ is equal to the number of columns of $A$. Mathematically, this can be written as: $\text{dim}(\text{ker }A)+\text{dim}(\text{im}A)=m$

Definition

The rank of a matrix $A$ is the dimension of the column space of $A$, denoted by $\text{rank}(A)$.

Definition

The nullity of a matrix $A$ is the dimension of the null space of $A$, denoted by $\text{nullity}(A)$.