The set

\[ \mathbb{H}^n = \{ (q_1, \ldots, q_n) \mid q_i \in \mathbb{H} \} \]

is a right vector space over the quaternions . That is, the scalar product from the right is defined by

\[ (q_1, \ldots, q_n) q := (q_1 q, \ldots, q_n q). \]
Remark

  • A map \( \varphi : \mathbb{H}^n \to \mathbb{H}^m \) is linear if and only if

    \[ \varphi(v + w) = \varphi(v) + \varphi(w), \quad \text{and} \quad \varphi(v q) = \varphi(v) q. \]

  • A linear map \( \varphi : \mathbb{H}^n \to \mathbb{H}^m \) is given by a matrix

    \[ A \in M^{n \times m}(\mathbb{H}), \]

    which is multiplied from the right.