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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 ....
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