# a5 - = Span(1 x x 2 deﬁned by φ f = f(1 ﬁnd g ∈ W...

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McGill University Math 247: Algebra 2 Assignment 5: due Wednesday, March 29, 2006 1. The trace of a square matrix A = [ a ij ] F n × n is the scalar tr(A) = n X i=1 a i i = a 11 + . . . + a nn . (a) Show that tr is linear and that tr(AB) = tr(BA). (b) If φ : F n × n F is linear and φ ( AB ) = φ ( BA ) for all A, B F n × n , show that φ = c tr for some scalar c . Hint: Show that the subspace spanned by the matrices of the form AB - BA has dimension n 2 - 1. 2. (a) Show that < ( x 1 , x 2 ) , ( y 1 , y 2 ) > = x 1 y 1 + 2 x 1 y 2 + 2 x 2 y 1 + 5 x 2 y 2 defines an inner product on R 2 . (b) Show that < ( x 1 , x 2 ) , ( y 1 , y 2 ) > = x 1 ¯ y 1 + ix 1 ¯ y 2 - ix 2 ¯ y 1 + 2 x 2 ¯ y 2 defines an inner product on C 2 . 3. Let V = C ([0 , 1]) with inner product < f, g > = R 1 0 f ( x ) g ( x ) dx. (a) Find the best approximation to f ( x ) = sin( πx ) by a function in Span(1 , x , x 2 ). (b) If φ is the linear form on
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Unformatted text preview: = Span(1 , x , x 2 ) deﬁned by φ ( f ) = f (1), ﬁnd g ∈ W such that φ ( f ) = < f,g > for all f ∈ W . 4. Let V = C n × n with the inner product < X,Y > = tr(X Y t ) and let T be the linear operator on V deﬁned by T ( X ) = • 1 i-i 1 ‚ X-X • 1 i-i 1 ‚ . (a) Show that T is self-adjoint. (b) Find an orthonormal basis of V consisting of eigenvectors of T . Hint: Use the fact that Im(T) ⊥ = Ker(T). (c) Find the matrix A of T with respect to the standard basis of V and ﬁnd a unitary matrix U such that U-1 AU is a diagonal matrix....
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