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

• HF08
In summary, the conversation is discussing the notation B(x,y) and (x,y) in the context of matrix multiplication. The question is asking for a proof that the norm of B(x,y) is equal to the norm of (x,y) for all (x,y) in R^2. The conversation also introduces the concept of vectors and ordered pairs. It suggests reading a textbook or asking a teacher for clarification on the notation. Finally, the original poster finds the problem easy once they understand the notation.
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

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 )

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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).

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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.

Solved

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

Regards,
HF08

## 1. What does ||B(x,y)|| mean in this equation?

The double vertical bars surrounding B(x,y) indicate the magnitude or length of the vector B(x,y). In this case, it represents the length of the vector (x,y) after it has been transformed by a rotation in R^2.

## 2. Can you explain what it means for two vectors to have equal magnitudes?

If two vectors have equal magnitudes, it means that they have the same length. In other words, they cover the same distance from their starting point to their end point. In the context of this equation, it means that the length of the vector (x,y) after the rotation is the same as the length of the original vector (x,y) before the rotation.

## 3. How does this equation relate to rotations in R^2?

This equation is a way to mathematically prove that rotations in R^2 preserve the length of vectors. In other words, the length of a vector after it has been rotated is the same as the length of the vector before it was rotated.

## 4. Is this equation only applicable to R^2 or can it be applied to other dimensions?

This equation is specifically for rotations in R^2, but the concept of preserving vector lengths can be applied to other dimensions as well. The equation would just need to be adjusted for the specific dimensions being studied.

## 5. How is this equation useful in the field of science?

This equation is useful in many areas of science, such as physics, engineering, and computer graphics. It allows us to mathematically prove the properties of rotations in R^2 and can be used to solve various problems involving vectors and rotations.

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