Help me fix all the problems that my professor have noted both in the paper and the email about quaternions.

Need help with my Mathematics question – I’m studying for my class.

The attached PDF is the annotated draft from my Professor. The Doc file is the original draft which you will need to work on and then send the revised version to me. I will give tips for quality work.
This revision work in my opinion is easy since there are only few parts of the paper need to be fixed.

Here is the email from my Professor.
I have attached the annotated draft. Here are my comments:
– There are some issues with line spacing.
– You repeated yourself a bit on the first page.
– I am happy that you added more connecting phrases between the sections.
– In the section on matrices you state that alpha = a+bi+cj+dk and alpha = aU+bI+cJ+dK. I don’t think you mean that these are literally equal. I think you mean that there is a function f that sends a quaternion alpha to a matrix f(alpha). You should also mention that this function preserves addition and multiplication: f(alpha+beta)=f(alpha)+f(beta) and f(alpha*beta)=f(alpha)*f(beta). If these formulas are not true then the representation would make no sense.
– You say to “recall” that for an imaginary quaternion v we have exp(v)= cos|v|+(v/|v|) sin|v|, but where are we supposed to recall this from? You didn’t prove it. To prove it, first define u=v/|v|, so that u^2=-1. Then we also have u^2=-1, u^3=-u, u^4=1, u^5=u, etc. Now plug v=u|v| into into the power series definition of the exponential and observe that
exp(v) = exp(u|v|) = sum_{k>=0} 1/k! u^k|v|^k = (power series for cos|v|) + u (power series for |v|).

Please make sure that all the anotations in red need to be revised! So do the points listed in the email above.

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