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MAT211_Lecture16[1]

MAT211_Lecture16[1] - Denition MAT211 Lecture 16 Orthogonal...

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MAT211 Lecture 16 Orthogonal transformations and orthogonal matrices Definition A linear transformation from R n to R n is called orthogonal if it preserves the length vectors. In symbols, ||T(x)||=||x|| for all x in R n . The matrix A of an orthogonal transformation is said to be an orthogonal matrix. Two examples: A rotation in R 2 and a reflexion in R n are orthogonal transformations. Questions: Are projections orthogonal transformations? What is the kernel of an orthogonal transformation? Example: Determine whether the matrices are orthogonal 1 1 1 1 -1 1 1 0 -2 0 1/ 2 1 1/ 2 1/ 3 Theorem An orthogonal transformation T in R n preserves angles; that is for each x and y in R n the angle between x and y equals the angle between T(x) and T(y). Question: If a transformation preserves angles, is it orthogonal?

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Theorem A linear transformation from R n to R n is orthgonal if and only if the vectors (T(e 1 ), T (e 2 ),.., T(e n )) form an orthonormal basis.
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