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# ef07k - MAS 3105 Final Exam and Key Prof S Hudson 1(10pts...

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April 26, 2007 Final Exam and Key Prof. S. Hudson 1) (10pts) Solve for X , given that AX + B = X and A = ± 2 2 0 2 ² B = ± - 1 - 2 - 3 0 ² 2) (10pts) Use the Gram Schmidt process to ﬁnd an orthonormal basis for R( A ). Then factor A = QR (orthogonal times upper triangular): A = ± - 1 3 1 5 ² 3) (15pts) a) Given that A is row equivalent to U , ﬁnd a basis for N ( A ). Check ! A = 1 - 2 1 1 2 - 1 3 0 2 - 2 0 1 1 3 4 1 2 5 13 5 U = 1 - 2 1 1 2 0 1 1 3 0 0 0 0 0 1 0 0 0 0 0 3b) Find a basis of the row space of A . Explain brieﬂy. 3c) Find a basis of the column space of A . Explain brieﬂy. 4) (10 pts) Show that any set of vectors that contains the zero vector must be L.D (de- pendent). Include the deﬁnition of LD in your proof. 5) (10pts) Let V = C [0 ] with the usual inner product h f,g i = 1 π R π - π f ( x ) g ( x ) dx . a) Compute || sin x || . b) Show that sin x cos x . Show all your work. 6) (5 pts) Choose ONE: a) Describe the eigshow utility used in several MATLAB problems in Ch6. b) Find two LI eigenvectors in C 2 for the L which rotates vectors 90 degrees CCW in R 2 . 7) (20 points) Answer True or False. Assume the matrices are 3x3, and that A is similar to B . If

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ef07k - MAS 3105 Final Exam and Key Prof S Hudson 1(10pts...

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