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 " tommy quaternion " MphLee Fellow Posts: 184 Threads: 19 Joined: May 2013 01/21/2021, 09:27 PM (This post was last modified: 01/21/2021, 10:59 PM by MphLee.) Hi, why you don't specify that $C*B=A$ in your list of identities? You forgot to add it or it is possible to derive it from the others? Also, you see it as a non-associative algebra over the reals or the complex? Trivial question (sorry but I'm new to linear algebra): being an algebra means that $\mathbb T$ is a $k$-vector space of dimension d=4 on which is defined a bilinear application $*:{\mathbb T} \times {\mathbb T}\to {\mathbb T}$ that is not associative... but it must be bilinear because we want it distributive: An element is of the form $T=x1+yA+zB+wC$. Let  $T,R,S,U\in{\mathbb T}$, the application being distributive means that $(T+R)*(S+U)=T*S+R*S+T*U+R*U$ i.e. every translation by a Tommy quaternion is a endomorphism of  the addition group; and we also want that for every scalar $\lambda$ $\lambda R*S= R*\lambda S=\lambda (R*S)$ i.e we want it to contain a copy of the base field, i.e multiplication by scalar  is multiplication by $\lambda1$ where $1\in \mathbb T$. So the previous means that the Tommy quaternions of the form $T=x1+0A+0B+0C$ commute and "associate" with everything. But this means that the multiplication is bilinear: every element should it should have a representation.. e.g. multiplication by $T$ has a 4x4 matrix $M_T$. Obviously if it is not associative we "should not" have $M_{T*S}=M_{T}M_{S}$ in general... Probably my understanding of non-associative alg. is so poor that I'm missing something obvious. MathStackExchange account:MphLee Fundamental Law $(\sigma+1)0=\sigma (\sigma+1)$ « Next Oldest | Next Newest »

 Messages In This Thread " tommy quaternion " - by tommy1729 - 01/18/2021, 11:03 PM RE: " tommy quaternion " - by marraco - 01/21/2021, 12:49 PM RE: " tommy quaternion " - by tommy1729 - 01/21/2021, 01:17 PM RE: " tommy quaternion " - by MphLee - 01/21/2021, 09:27 PM RE: " tommy quaternion " - by tommy1729 - 01/21/2021, 11:47 PM RE: " tommy quaternion " - by MphLee - 01/22/2021, 12:03 AM RE: " tommy quaternion " - by tommy1729 - 01/22/2021, 12:15 AM RE: " tommy quaternion " - by tommy1729 - 01/22/2021, 12:10 AM RE: " tommy quaternion " - by MphLee - 01/22/2021, 12:41 AM RE: " tommy quaternion " - by tommy1729 - 01/23/2021, 01:17 PM RE: " tommy quaternion " - by marraco - 01/24/2021, 06:53 AM RE: " tommy quaternion " - by tommy1729 - 02/12/2021, 11:17 PM RE: " tommy quaternion " - by JmsNxn - 02/14/2021, 02:24 AM RE: " tommy quaternion " - by tommy1729 - 03/23/2021, 01:21 PM RE: " tommy quaternion " - by tommy1729 - 09/16/2021, 11:34 PM

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