This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 Math 1b Practical Isometries March 4, 2009 We have explained that a linear transformation of R n to itself is an isometry when its matrix is an orthogonal matrix. [The converse is also true.] An example of an isometry is the reection through a subspace U of R n . The reection of x through U is defined as follows: if x = u + w with u U and w U , then the reection re U ( x ) is simply u w . It follows that re U ( x ) = 2 proj U ( x ) x . The proofs of the following two theorems are short, but understanding and reading them will be dicult. We will not do all details in class. Theorem 1. Every n by n orthogonal matrix Q is the product of at most n matrices of reflections through hyperplanes. Proof: Recall that a hyperplane H is an ( n 1)dimensional subspace. We will use R H to denote the matrix [re H ] of the reection through H . So R H is an orthogonal matrix and R 2 H = I ....
View
Full
Document
This note was uploaded on 09/25/2010 for the course MATH 1bPRAC taught by Professor Wilson during the Winter '09 term at Caltech.
 Winter '09
 Wilson
 Math, Linear Algebra, Algebra

Click to edit the document details