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take_home7 - 3 Wlth Initial conditions y = gm 3 = ya 3 =...

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Unformatted text preview: . . . . . . 3 Wlth Initial conditions y = gm, 3% = ya, 3:!- = 93', .H, “—1 I I I if??- = 93—1, when :t: = 3:0. (Hint: convert It Into a system of first order equations and apply previous problem.) 1'3] Let R“ be equipped with the Euclidean norm llsll = [216]“ Let MHUE} be the set of rs x n. matrices. and note that MfiR) is a rig-dimensional real vector space. Show that the formula {Ll-(A) = Ell-113-""{ll14l:-“Elll III: 1: 1} defines a norm on MAR}. Determine how to effectively compute ill-{A}- [Hints: You may use without proof the following facts from linear algebra: There is an algorithm for writing any matrix A in the form A 2 UP. where U is an orthogonal matrix and P is a symmetric matrix with nonnegative eigenvalues. Furthermore. there is an orthogonal change of basis to diagonalize a symmetric matrix.) ED] Show that the real projective plane RP“ = Smfx # —x is homeomorphic to the space of n K n symmetric matrices of rank 1. ...
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