Definition 2

The transpose of a matrix $M = (m_{\iota \kappa})_{(\iota, \kappa) \in I \times K}$, denoted by $^tM$, is the matrix $(m'_{\kappa \iota})_{(\kappa, \iota) \in K \times I}$ over $H$ given by $m'_{\kappa \iota} = m_{\iota \kappa}$ for all $(\iota, \kappa) \in K \times L$.

It follows from this definition that the transpose of a matrix of type $(I, K)$ is a matrix of type $(K, I)$ and that

(1)