assig6 - MATH 245 Linear Algebra 2, Assignment 6 Due Fri...

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MATH 245 Linear Algebra 2, Assignment 6 Due Fri Dec 3 1: (a) For the quadratic curve 7 x 2 + 8 xy + y 2 + 5 = 0, find the coordinates of each vertex, find the equation of each asymptote, and sketch the curve. (b) For the real quadratic form K ( x,y,z ) = 3 x 2 + ay 2 + bz 2 - 6 xy +2 xz - 4 yz , sketch the set of points ( a,b ) for which K is positive-definite. 2: Let U and V be non-trivial subspaces of R n with U V = { 0 } . Recall that angle( U,V ) = min ± angle( u,v ) ² ² 0 6 = u U, 0 6 = v V ³ . (a) Show that angle( U,V ) = cos - 1 ( σ ) where σ is the largest singular value of the linear map P : U V given by P ( x ) = Proj V ( x ). (b) Let u 1 = 1 1 - 1 1 , u 2 = 2 1 - 1 2 , v 1 = 1 1 0 - 1 and v 2 = 2 1 1 0 . Let U = Span { u 1 ,u 2 } and V = Span { v 1 ,v 2 } . Find angle( U,V ). 3: Let F = Z 7 , the field of integers modulo 7. (a) Let
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