Linear Algebra Help

[daveollie]

I'm not too sure what the question is asking me to do, some advice would be much appreciated, thank you.

 

[HallsofIvy]

It looks to me like it is asking you to write down the transformation, in terms of a matrix or "x'= ", "y'= " form, of rotations through angles from 0 to \(\displaystyle \frac{n-1}{n}(2\pi)\).

The matrix form for a rotation through angle \(\displaystyle \theta\) is
\(\displaystyle \begin{bmatrix} x' \\ y'\end{bmatrix}= \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}\)

Or, equivalently, \(\displaystyle x'= x cos(\theta)- y sin(\theta)\), \(\displaystyle y'= x sin(\theta)+ y cos(\theta)\).

 

[Ishuda]

View attachment 4614I'm not too sure what the question is asking me to do, some advice would be much appreciated, thank you.

Do you know the general formula for a rotation? If there is no stretching or contraction then it is
u = a x + b y
v =-b x + a y
where
\(\displaystyle a^2 + b^2 = 1\)

As a hint: \(\displaystyle cos^2(\theta) + sin^2(\theta) = 1\)

 

[daveollie]

It looks to me like it is asking you to write down the transformation, in terms of a matrix or "x'= ", "y'= " form, of rotations through angles from 0 to \(\displaystyle \frac{n-1}{n}(2\pi)\).

The matrix form for a rotation through angle \(\displaystyle \theta\) is
\(\displaystyle \begin{bmatrix} x' \\ y'\end{bmatrix}= \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}\)

Or, equivalently, \(\displaystyle x'= x cos(\theta)- y sin(\theta)\), \(\displaystyle y'= x sin(\theta)+ y cos(\theta)\).

Do you know the general formula for a rotation? If there is no stretching or contraction then it is
u = a x + b y
v =-b x + a y
where
\(\displaystyle a^2 + b^2 = 1\)

As a hint: \(\displaystyle cos^2(\theta) + sin^2(\theta) = 1\)

Thanks guys, maybe it would have made more sense to you if I had given the full question, here it is, I am really confused by this to the point where I don't understand properly what I am being asked to do. Thank you.

 

[Ishuda]

Thanks guys, maybe it would have made more sense to you if I had given the full question, here it is, I am really confused by this to the point where I don't understand properly what I am being asked to do. Thank you....

Sounds like you are being asked to develop those equations. That is rotate a point clockwise in the x, y plane anti-clockwise by \(\displaystyle \theta\) and develope the equations for the new point (u, v) [or as HallsofIvy has it (x', y')] in terms of the co-ordinates (x,y).

BTW: I think HallsofIvy has it as a clockwise transformation which, if so, would mean you would replace his \(\displaystyle \theta\) by -\(\displaystyle \theta\) to arrive at the matrix you have in the problem. For my notation, b = sin(\(\displaystyle \theta\)) and a = cos(\(\displaystyle \theta\))

 

[stapel]

I've numbered for ease of reference:

Define the following:

i) A rotation \(\displaystyle D_r\) as an anticlockwise rotation in the x-y plane through the angle \(\displaystyle 360r/n, \, r\, =\, 0\) to \(\displaystyle n\, -\, 1, r\, \in\, \mathbb{N}.\)

ii) The set \(\displaystyle D(n)\, =\, \{D_0,\, D_1,\, D_2,\, ...,\, D_{n-1}\}i\)

Show the following:

iii) \(\displaystyle Dr\) can be represented as a matrix \(\displaystyle \left(\begin{array}{cc}\cos\left(360r/n\right)&\sin\left(360r/n\right)\\-\sin\left(360r/n\right)&\cos\left(360r/n\right)\end{array}\right),\, r\, =\, 0\) to \(\displaystyle n\, -\, 1,\, r\, \in\, \mathbb{N},\, r\)

What is the meaning of the "i" after the listing of elements in D(n)? Is "Dr" meant to be "D_r"? What is the meaning of the "r" at the end of the statement in (iii)?

Thank you!

 

[daveollie]

I've numbered for ease of reference:


What is the meaning of the "i" after the listing of elements in D(n)? Is "Dr" meant to be "D_r"? What is the meaning of the "r" at the end of the statement in (iii)?

Thank you!

I don't know, I have just snipped the picture and posted it as it is on the sheet. I'll ask in tomorrow's lecture and see if I can find out.

 

[daveollie]

I've numbered for ease of reference:


What is the meaning of the "i" after the listing of elements in D(n)? Is "Dr" meant to be "D_r"? What is the meaning of the "r" at the end of the statement in (iii)?

Thank you!

you were right in all of you're assumptions there, i shouldn't be there, neither should the r, and the other r is a subscript of D. Our lecturer has thanked me for pointing his typo's out, I now feel like a smart arse haha, but if you can give me any new help now that I've cleared up the little annoying things, I'd appreciate it loads