The set of linear functionals \(\mathbb{R}^{n m} ightarrow \mathbb{R}\) is called the dual vector space; it has dimension \(m n\). Show that the column and row totals \(Y \mapsto Y_{. r}\) and \(Y \mapsto Y_{i \text {. are }}\) linear functionals, and that they are linearly independent. Show that the subspace spanned by \(\left\{Y_{.1}, \ldots, Y_{. m}ight\}\) is closed with respect to row and column permutations. What is its dimension? Show that the subspace spanned by \(\bar{Y}_{. .}\), and the subspace spanned by \(\left\{\bar{Y}_{.1}-\bar{Y}_{. .}, \ldots, \bar{Y}_{. m}-\bar{Y}_{. .}ight\}\)are both closed with respect to row and column permutations. What are their dimensions?