These functions return an antiassociative algebra element with the
specified coefficients. Given a numeric vector v with elements
\(v_1,v_2,\ldots, v_n\) then
linear1(v) returns \(v_1\mathbf{a} + v_2\mathbf{b}+\cdots+
v_n\mathbf{L_n}\), where \(\mathbf{L_n}\) is the
\(n^\mathrm{th}\) letter of the alphabet. Similarly,
linear2(v) returns \(v_1\mathbf{a}\mathbf{a}+\cdots+
v_n\mathbf{L_n}\mathbf{L_n}\), and linear3(v) returns
\(v_1(\mathbf{a}\mathbf{a})\mathbf{a}+\cdots+
v_n(\mathbf{L_n}\mathbf{L_n})\mathbf{L_n}\). They are linear in
the sense that
$$
f(\alpha\mathbf{x}+\beta\mathbf{y})=
\alpha f(\mathbf{x})+\beta f(\mathbf{y})$$
where \(\alpha,\beta\in\mathbb{R}\) and
\(\mathbf{x},\mathbf{y}\in\mathbb{R}^n\).