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Linear Algebra Workbook - Florida Atlantic University

Let H be the quaternion algebra. Let x = a+bi+cj+dk ∈ H. Define its conjugate to be the quaternion x = a − bi− cj −dk. (i) Show that xx = xx = (a2 +b2 + c2 + d2). (ii) Show that H is a division algebra, i.e., xy = 0 =⇒ x = 0 or y = 0. (ii) Let q be a nonzero quaternion. Show that {q, qi, qj, qk} is a basis of H.