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Let us denote the set of all Hankel matrices by H. Let $AB鈭�H$ and k be any scalar.

$$\begin{gathered} (A+B{)}_{ij}=A_{ij}+B_{ij} \\ =A_{i+1,j-1}+B_{i+1,j-1} \\ =(A+B{)}_{i+1,j-1} \end{gathered}$$

Thus,$AB鈭�H$.

$$\begin{gathered} {\left(kA\right)}_{ij}=k{A}_{ij} \\ =k{A}_{i+1,j-1} \\ ={\left(kA\right)}_{i+1,j-1} \end{gathered}$$

Thus, $kA鈧�H$.

Hence, H is a subspace of ${\mathrm{R}}^{\mathrm{n}脳\mathrm{n}}$.

Now, in order to find out the dimension of H. it needs to write-down an arbitrary element.

$$\begin{gathered} H=\left[\begin{array}{cc}{a}_{11}& a_{12}....a_{1n}\\ a_{21}& a_{22}....a_{2n}\\ .& ......\\ .& .....\\ a_{n1}& a_{n2}...a_{nn}\end{array}\right]:\left[a_{ij}=a_{i+j,j-1},i=1,2,...n-1,j=2,...n\right] \\ =\left[\begin{array}{cccc}{a}_{11}& a_{12}& & a_{13}........a_{1n}\\ a_{12}& & & a_{13}............a_{2n}\\ a_{13}& & & ...................a_{3n}\\ .& & & .\\ ..& & & a_{n-1,n}{a}_{n-1},n\\ a_{1n}{a}_{2n}& ..a_{n-2,n}& a_{n-1,n}& a_{nm}\end{array}\right]a_{ij}鈭�R \end{gathered}$$

Consider the diagram below.

Observe that, n-1 independent elements are there in the upper block and n-1 independent elements are there in the lower block and one in the diagonal.

Therefore, the total number of independent elements are

$(n-1)+1+(n-1)=2n-1$

Hence, the dimension of n脳nHankel matrices is 2n-1 .