Exercise 28

Prove (where $A$ is an $n\!\times\!n$ matrix and so defines a transformation of any $n$-dimensional space $V$ with respect to $B,B$ where $B$ is a basis) that $\dim(\mathscr{R}(A)\cap\mathscr{N}(A))=\dim(\mathscr{R}(A))-\dim(\mathscr{R}(A^{2}))$. Conclude

• $\mathscr{N}(A)\subset\mathscr{R}(A)$ iff $\dim(\mathscr{N}(A))=\dim(\mathscr{R}(A))-\dim(\mathscr{R}(A^{2}))$;

• $\mathscr{R}(A)\subseteq\mathscr{N}(A)$ iff $A^{2}=0$;

• $\mathscr{R}(A)=\mathscr{N}(A)$ iff $A^{2}=0$ and $\dim(\mathscr{N}(A))=\dim(\mathscr{R}(A))$ ;

• $\dim(\mathscr{R}(A)\cap\mathscr{N}(A))=0$ iff $\dim(\mathscr{R}(A))=\dim(\mathscr{R}(A^{2}))$ ;

• (Requires the Direct Sum subsection, which is optional.) $V=\mathscr{R}(A)\oplus\mathscr{N}(A)$ iff $\dim(\mathscr{R}(A))=\dim(\mathscr{R}(A^{2}))$.