Bijectivity

A map $f:A\to B$ (i.e., a function or linear map) is bijective if it is both injective and surjective. If this is the case, there is a 1-1 correspondence between the domain and codomain sets.

Linear transformations ($n\times m$) are bijective if they have the following characteristics:

• $\text{rref}(A)=I_n$, i.e., there is a pivot in each row and column
• $\text{rank}(A)=m=n$, i.e., it is square
• The matrix is invertible
• The linear system $A\vec{x}=\vec{b}$ has a unique solution $\vec{x}$ for all $\vec{b}$ in ${\mathbb{R}}^n$
• $\text{im}(A)={\mathbb{R}}^n$
• $\text{ker}(A)=\vec{0}$
• $\text{det}(A)\ne 0$