 This duality is easily visualized in the concrete approach by viewing linear transformations from \(U\) to \(V\) as \(m\times n\) matrices. The duality simply transposes the matrices while leaving the machinery of matrix multiplication itself unchanged. It is then immediate that this operation is an involution that reverses maps—the \(m\times n\) matrix linearly transforming an \(n\)-dimensional space \(U\) to an \(m\)-dimensional one \(V\) transposes to an \(n\times m\) matrix linearly transforming the \(m\)-dimensional space \(V^*\) to the \(n\)-dimensional space \(U^*\).