LADR 3.B.5
This is taken from Sheldon Axler's magisterial book, aptly named “Linear Algebra Done Right”. The exercise (3.B.5 on p. 67) asks for an example of a linear map $T: \mathbb{R}^4 \rightarrow \mathbb{R}^4$ such that $\mathrm{range}(T)=N(T)$ where $N(T)$ is the null space of $T$. We know that a matrix $A$, \begin{equation} A = \begin{bmatrix} A_1 & A_2 & A_3 & A_4 \end{bmatrix} \end{equation} with $A_1$, $A_2$ two independent vectors and $A_3=aA_1+bA_2$, $A_4=cA_1+dA_2$ has a range equal to $\mathrm{Span}\{A_1,A_2\}$. $N(A)$ contains vectors $x$ with the property, \begin{equation} x_1 A_1 + x_2 A_2 + x_3 A_3 + x_4 A_4 = 0 \end{equation} or, \begin{equation} (x_1+ax_3+cx_4) A_1 + (x_2+bx_3+dx_4) A_2 = 0 \label{null_cond} \end{equation} Since $A_1$, $A_2$ are independent vectors ($\ref{null_cond}$) holds only if, \begin{align} x_1 & = - a x_3 - c x_4 \\ x_2 & = - b x_3 - d x_4 \end{align} Therefore any $x...