[NEED HELP ] Linear Systems Calculations

[ivanyo]

Hi guys/gals,

First post here - in need of some desperate last minute help. Today is my last day to study and I will have my final exam tommorow. I've been doing past papers all day, and there a few questions I have no idea/never seen before in my course material.

Part (c) I have no idea what to do. I know I need to convert the 4sqrt2 into Cartesian, but I have no idea how...

Part (d) I'm very close to getting, but I end up getting -cos^2(theta) + (-1) sin^2(theta) = 0

Part (e) is related to part (d), so I'm not too sure at the moment.

I know that there is the rule that when there is a matrix A, A1v1=A2v2. However, these eigenvalues are imaginary, so how do I calculate the eigenvectors?

In a)i) of this question, I calculated -1/2 and -1 as the values of M (i.e. when you let Un=kM^n). I just want to double check these are correct, and also would like somebody to guide me in part ii). I have no idea how to calculate this using the results of part i)MANY MANY MANY thanks to whoever is able to help me out here. I think I may have already failed 1 subject this semster, and do not want to make it 2.

Cheers,

EDIT- extremely sorry for the image size, I don't know how to re-frame it. it's a photo from my phone haha

 

[rock.freak667]

For the resizing thing, if you upload to imageshack there is an option for it.

First image:

c)the 4√2 is just a constant, so when you put e^iπ/4 into cartesian form, you would multiply it by 4√2.

d) If you consider z₁z₂ in polar form you will get r² right? So just expand z₁z₂ in cartesian form and then put that equal to r² [z₁ is the z in the question and z₂ is its conjugate]

e) Well you want to find sin5θ, so consider expanding (cosθ+isinθ)⁵ (which is equak to cos5θ+isin5θ by De Moivre's theorem) and equate real and imaginary parts.

I am not too well versed on the topics in the second and third images, so I can't really help you there.

 

[ivanyo]

Thanks for the response. I think I get it a little better now.

Any takers for 2nd+3rd image?