- #1

- 10

- 0

## Homework Statement

Let L: R

^{2}→ R

^{n}be a linear mapping. We call L a

*similarity*if L stretches all vectors by the same factor. That is, for some δ

_{L}, independent of v,

|L(v)| = δ

_{L}* |v|

To check that |L(v)| = δ

_{L}* |v| for all vectors v in principle involves an infinite number of calculations. Therefore a reduction to a finite number of checks can be useful for applications. Here is one such reduction.

Suppose that for the standard orthonormal basis {e

_{1}, e

_{2}} of R

^{2}, we have that L(e

_{1}) and L(e

_{2}) are orthogonal and have the same length. Show that L is a similarity. (What is δ

_{L}?)

## Homework Equations

Perhaps that orthogonal vectors have dot product zero? I'm not sure otherwise.

## The Attempt at a Solution

So far I've been able to show that |L(e

_{1})| = |L(e

_{2})| = δ

_{L}on the assumption that L is a similarity, but that doesn't really help that much. I'm not sure where to go beyond this.