5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL …

A linear transformation T from Rn to Rn is called orthogonal if it preserves the length of vectors: kT(~x)k = k~xk, for all ~x in Rn. If T(~x) = A~x is an orthogonal transformation, we say that A is an orthogonal matrix. 1

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8.2 Quadratic Forms Example 1 - NCU

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8.2 Quadratic Forms Example 1 Consider the function q(x1;x2)=8x21 4x1x2 +5x22 Determine whether q(0;0) is the global mini-mum. Solution based on matrix technique Rewrite

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21. Orthonormal Bases

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Then as a linear transformation, P i w iw T i = I n xes every vector, and thus must be the identity I n. De nition A matrix Pis orthogonal if P 1 = PT. Then to summarize, Theorem. A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. i.e. P 1 = PT: Example Consider R3 with the orthonormal basis S= 8 >> < >>: u 1 = 0 ...

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Linear Algebra Problems - Penn Math

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g) The linear transformation T A: Rn!Rn de ned by Ais onto. h) The rank of Ais n. i) The adjoint, A, is invertible. j) detA6= 0. 14. Call a subset S of a vector space V a spanning set if Span(S) = V. Suppose that T: V !W is a linear map of vector spaces. a) Prove that a linear map T is 1-1 if and only if T sends linearly independent sets

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Linear Transformation Exercises

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Linear Transformation Exercises Olena Bormashenko December 12, 2011 1. Determine whether the following functions are linear transformations. If they are, prove it; if not, provide a counterexample to one of the properties: (a) T : R2!R2, with T x y = x+ y y Solution: This IS a linear transformation. Let’s check the properties:

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