Prove ||B(x,y)|| = ||(x,y)|| for all x,y in R^2 (Rotations in R^2)

[HF08]

B = [cos$$\theta$$ -sin$$\theta$$]
...[sin$$\theta$$ cos$$\theta$$]

for some $$\theta$$ in R$$^{2}$$.


(a) Prove that || B(x,y) || = || (x,y) || for all (x,y)$$\in$$R$$^{2}$$

Question: What does B(x,y) and (x,y) notation mean?
I have a result that says

Let B=[b$$_{ij}$$] be an mxn matrix whose entries
are real numbers and let e$$_{1}$$,...,e$$_{n}$$ represent the usual basis of R^n. If T(x) = Bx, x$$\in$$R^n , then T is a linear function from R^n to R^m and T(e$$_{j}$$)=(b$$_{1j}$$,b$$_{2j}$$,...,b$$_{mj}$$, j = 1,2,...n


Warning: Superscripts are not superscipts. They are supposed to be SUBSCRIPTS. Sigh.

Can I use this?

1. I am very new to this material
2. I am stuck with the notation.
3. Please answer my first question carefully. I can't answer the question unless I know what they are asking. :)

Please help me. Thank You,
HF08

 

[Mystic998]

Not to be a jerk, but I'm not sure you should be doing problems like this if you don't even know what a vector or at least an ordered pair is. Regardless, here's a short introduction to vectors via Wikipedia.

http://en.wikipedia.org/wiki/Vector_(spatial )

 

[HallsofIvy]

Better yet, read your textbook. As a "last resort"(!) ask your teacher what those things mean. Surely whoever gave you that problem was assuming you already knew that B(x,y) means to multiply the matrix B by the (column) vector (x, y).

 

[HF08]

Ah...

I know what a vector and ordered pair is. So what they are really saying is this:
Bx=x, right? If so, that makes a lot more since then the (x,y) notation to me.

 

[HF08]

Solved

This was easy! Thanks for your kind replies. My problem was the notation, after that, it just follows very quickly.

Regards,
HF08