You know, I think that's pretty much what I am being asked to show. I appreciate your insights! So basically (and please excuse the poor notation, I'm on my phone):
(H,\otimes,°,\oplus, =)~(M, ×, •, +, =)
Okay, so an isomorphism is a bijectivity from one structure to another which preserves the characteristics of the structures. So if quaternions possesses addition, scalar and quaternion multiplication I must show that this structure is isomorphic to the addition, scalar, and matrix...
The more I look at this problem the less I understand what I am supposed to be doing. Without giving me any answers what I need is for someone to walk me through the steps to the solution. I need an understanding of what I'm trying to solve here.
I just started learning about morphisms and I came across a problem that totally stumps me. Here goes:
Show that the algebra of quaternions is isomorphic to the algebra of matrices of the form:
\begin{pmatrix}
\alpha & \beta \\
-\bar{\beta} & \bar{\alpha} \end{pmatrix}
where α,β\inℂ...
Arc length of y=ln((e^(x)+1)/(e^(x)-1)) on [a,b]
Using L=\int\sqrt{1+(y')^2}dx on [a,b]
I am having difficulties differentiating y and plugging the results back into get a useful integral. So far I have y'=2e^(x)/(e^(2x)-1)