Chapter 5.3, Problem 71E

An $n\times n$ matrix A is said to be a Hankel, matrix (namedafter the German mathematician Hermann Hankel, 1839-1873) if $a_{ij}=a_{i+1,j-1}$ forall $i=1,\mathrm{...},n-1$ and all $j=2,\mathrm{...},n$ , meaning that A has constant positive sloping diagonals. For example, a $4\times 4$ Hankelmatrix is of the form $A=\left[\begin{array}{cccc}a& b& c& d\\ b& c& d& e\\ c& d& e& f\\ d& e& f& g\end{array}\right]$ .

Show that the $n\times n$ Hankel matrices form a subspace of ${ℝ}^{n\times n}$ . Find the dimension of this space.