Consider the standard matrix A representing the linear transformationT(\ ) = v x x from

Chapter 8, Problem 22

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Consider the standard matrix A representing the linear transformationT(\ ) = v x x from to Jwhere r is a given nonzero vector in IR3. a. Use the geometrical interpretation of the cross product to find an orthogonal projection T\ onto a plane, a scaling 72. and a rotation 7j about a line such that T{x) = 73 (72(7'| (.v))). for all x in !R-\ Describe the transformations T\, T2 and 73 as precisely as you can: For T\ give the plane onto which we project, for 72 find the scaling factor, and for Ty give the line about which we rotate and the angle of rotation. All of these answers, except for the angle of rotation, will be in terms of the given vector 5. Now let Aj, A2, and Ay be the standard matrices of these transformations T\. 72 and 73, respectively. (You are not asked to find these matrices.) Explain how you can use the factorization A = AyA2A \ to write a polar decomposition A = QS of A. Express the matrices Q and S in terms of A \, A2, and Ay. See Exercise 20. Find the AyA2A\ and QS factorizations discussed in part (a) in the caseO' 2 0

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