- #1
math222
- 10
- 0
Homework Statement
Let L: R2 → Rn be a linear mapping. We call L a similarity if L stretches all vectors by the same factor. That is, for some δL, independent of v,
|L(v)| = δL * |v|
To check that |L(v)| = δL * |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 {e1, e2} of R2, we have that L(e1) and L(e2) are orthogonal and have the same length. Show that L is a similarity. (What is δL?)
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(e1)| = |L(e2)| = δL 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.