math54-unknown-mt1-Evans-exam

math54-unknown-mt1-Evans-exam - A corresponding to the...

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MATH 54 – OLD MIDTERM #2 Problem #1 (a). Find the eigenvalues of the matrix · 41 14 ¸ . (b). Compute the Wronskian of the functions y 1 ( x )= e x cos x, y 2 ( x )= e x sin x. Problem #2. Find the determinant of the matrix 1012 1140 1102 1101 . Problem #3. Apply the Gram–Schmidt process to convert the vectors v 1 =(1 , 2 , 1) , v 2 =(1 , 1 , 1) , v 3 =(1 , 2 , 1) into an orthonormal basis of R 3 . Problem #4. Let A be a real n × n symmetric matrix. Prove that if v 1 , v 2 are eigenvectors of
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Unformatted text preview: A corresponding to the distinct eigenvalues 1 , 2 , then v 1 , v 2 are orthogonal. Problem #5. Let A be a real n n matrix and consider the symmetric matrix B = A T A . Show that if is an eigenvalue of B , then . (Hint: Since is an eigenvalue, we have B v = v for some eigenvector v 6 = . Now use the dot product.)...
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