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s11_mthsc853_hw09

# s11_mthsc853_hw09 - 1 Deﬁne an inner product on X by f,g...

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MthSc 853 (Linear Algebra) Dr. Macauley HW 9 Due Friday, April 1st, 2011 (1) Consider the following matrix: M n = 0 - a 0 I n - 1 - a n - 1 where a n = a 1 a 2 . . . a n . (a) Show that the characteristic polynomial of M n is P M n ( t ) = t n + a n - 1 t n - 1 + · · · + a 1 t + a 0 . Here, I n - 1 denotes the ( n - 1) × ( n - 1) identity matrix. (b) Is P M n ( t ) also the minimal polynomial? Prove or disprove. (c) Now, let X be a 4 dimensional vector space over R with basis { x 1 , x 2 , x 3 , x 4 } and let T : X X be a linear map such that T ( x 1 ) = x 2 , T ( x 2 ) = x 3 , T ( x 3 ) = x 4 , T ( x 4 ) = - x 1 - 4 x 2 - 6 x 3 - 4 x 4 . Is T diagonalizable over C ? (2) Prove that in a real Euclidean space, || x || = max { ( x, y ): || y || = 1 } . (3) Let f and g be continuous functions on the interval [0 , 1]. Prove the following inequalities. (a) Z 1 0 f ( t ) g ( t ) dt 2 Z 1 0 f ( t ) 2 dt Z 1 0 g ( t ) 2 dt (b) Z 1 0 ( f ( t ) + g ( t )) 2 dt 1 / 2 Z 1 0 f ( t ) 2 dt 1 / 2 + Z 1 0 g ( t ) 2 dt 1 / 2 . (4) Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R 4 spanned by y 1 = (1 , 2 , 1 , 1), y 2 = (1 , - 1 , 0 , 2) and y 3 = (2 , 0 , 1 , 1). (5) Let X be the vector space of all continuous real-valued functions on [0
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Unformatted text preview: , 1]. Deﬁne an inner product on X by ( f,g ) = Z 1 f ( t ) g ( t ) dt. Let Y be the subspace of X spanned by f ,f 1 ,f 2 ,f 3 , where f k ( x ) = x k . Find an orthonor-mal basis for Y . (6) Let Y be a subspace of a Euclidean space X , and P Y : X → X the orthogonal projection onto Y . Prove that P * Y = P Y . (7) Show that a matrix M is orthogonal iﬀ its column vectors form an orthonormal set. (8) Let X be an n-dimensional real Euclidean space, and A : X → X a linear map. Deﬁne the map f : X → X by f ( x,y ) = x T Ay . Give (with proof) necessary and suﬀcient conditions on A for f to be an inner product on X ....
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