Let $A\in {\mathbb{R}}^{n\times n}$ be a square matrix with real entries. Then, the following statements are equivalent $⟺$. In other words, for a given matrix $A$, the statements are either all true or all false.
13. The determinant of $A$ is not zero: set $(A)\ne 0$.
14. The columns of $A$ form a basis for ${\mathbb{R}}^n$.
15. The column space of $A$ is ${\mathbb{R}}^n:\text{Col}(A)={\mathbb{R}}^n$.
16. The dimension of the column space of $A$ is $n:\text{dim}(\text{Col}(A))=n$
17. The rank of $A$ is $n:\text{rank}(A)=n$.
18. The dull space of $A$ is $\{\overrightarrow{0}\}:\text{Null}(A)=\{\overrightarrow{0}\}$
19. The dimension of the null space of $A$ is $0:\text{dim}(\text{Null}(A))=0$.
20. The orthogonal complement of the column space of $A$ is $\{\overrightarrow{0}\}$. We can write this as

21. The orthogonal complement of the null space of $A$ is ${\mathbb{R}}^n$: We write this

22. The row space of $A$ is ${\mathbb{R}}^n:\text{Row}(A)={\mathbb{R}}^n$.