# Matrix Transformations

1. Apr 12, 2007

### 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

2. Apr 12, 2007

### Integral

Staff Emeritus
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.

3. Apr 12, 2007

### 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?

4. Apr 12, 2007

### Tom Mattson

Staff Emeritus
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.

5. Apr 12, 2007

### robphy

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

6. Apr 12, 2007

### Mystic998

I feel a collective groan would be appropriate here.