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

1. Feb 24, 2008

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.

HF08

2. Feb 25, 2008

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)

3. Feb 25, 2008

HallsofIvy

Staff Emeritus
Better yet, read your text book. 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).

Last edited: Feb 25, 2008
4. Feb 25, 2008

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 alot more since then the (x,y) notation to me.

5. Feb 26, 2008

HF08

Solved

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

Regards,
HF08