Preserving Vector Length: Exploring Matrix Transformations in R^2

[SNOOTCHIEBOOCHEE]

Find all matrix transformations f:R^2 -----> R^2 which leave the length of vectors in the plane unchanged



Thats R as in the set of all real numbers R.


The only possible transformations i could possibly think of that would not change the length is rotation, other than that i am completley lost

 

[Integral]

There is another type which does not effect length.

I would start by looking at a general transform matrix and learning which elements effect lenght, how would you arrange it so the length remains unchanged. You will then need to do some form of proof.

 

[Mystic998]

Just my random thoughts in trying to solve the problem: Wouldn't that be equivalent to saying t(AX) * AX = t(X) * X, where t(X) is the transpose of X?

 

[quantumdude]

The only possible transformations i could possibly think of that would not change the length is rotation, other than that i am completley lost

Try to do the following in order.

1.) Given a vector $\vec{v}$ in $\mathbb{R}^2$, write down an expression for its length.

2.) Now transform the vector by multiplying it by a 2x2 matrix $A$:

$\vec{v} \rightarrow \vec{v}^{\prime}=A\vec{v}$.

3.) Write down an expression for the length of $\vec{v}^{\prime}$ in terms of $\vec{v}$ and $A$.

4.) Given that the lengths of $\vec{v}$ and $\vec{v}^{\prime}$ must be equal, deduce a condition for $A^TA$.

Try those steps and let us know where you get stuck.

 

[robphy]

After thinking about it from all angles, reflect on the problem a little more.

 

[Mystic998]

After thinking about it from all angles, reflect on the problem a little more.

I feel a collective groan would be appropriate here.