assign5-sol

assign5-sol - Math 235 Assignment 5 Due 9:15am, Wednesday...

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Unformatted text preview: Math 235 Assignment 5 Due 9:15am, Wednesday Feb 28, 2007. 1. From the Text 5.3 #31. Construct a nonzero 2 2 matrix that is invertible but not diag- onalizable. Solution. Consider a matrix 1 1 1 . The determinant of this matrix is one, implying this matrix is invertible. It is easy to see that the matrix has only one eigenvalue = 1 with algebraic multiplicity two. The eigenspace corresponding to the eigenvalue is spanned by 1 , thus geometric multiplicity of is one which is less than algebraic multiplicity of . Therefore the matrix is not diagonalizable. #32. Construct a nondiagonal 2 2 matrix that is diagonalizable but not invertible. Solution. Consider 1 1 1 1 . The determinant of the matrix is zero, implying this matrix is not invertible. The characteristic equation of the matrix is given by ( - 2) = 0 . Thus there are two distinct eigenvalues, 1 = 0 and 2 = 2 . Since the dimension of the matrix is also two, then the matrix is diagonalizable. 5.4 #10. Define T : P 3- R 4 by T ( p ) = p (- 3) p (- 1) p (1) p (3) . a. Show that T is a linear transformation. b. Find the matrix for T relative to the basis { 1 , t, t 2 , t 3 } for P 3 and the standard basis for R 4 . Solution. a. We need to show that T ( c p + d q ) = cT ( p ) + dT ( q ) , where c and d are constants. 1 T ( c p + d q ) = ( c p + d q )(- 3) ( c p + d q )(- 1) ( c p + d q )(1) ( c p + d q )(3) = c ( p (- 3)) + d ( q (- 3)) c ( p (- 1)) + d ( q (- 1)) c ( p (1)) + d ( q (1)) c ( p (3)) + d ( q (3)) = c p (- 3) p (- 1) p (1) p (3) + d q (- 3) q (- 1) q (1) q (3) = cT ( p ) + dT ( q ) b. Let B = { 1 , t, t 2 , t 3 . } Then p ( t ) = a + bt + ct 2 + dt 3 = a b c d B . We want to find a matrix A such that A a b c d B = p (- 3) p (- 1) p (1) p (3) = a + b (- 3) + c (- 3) 2 + d (- 3) 3 a + b (- 1) + c (- 1) 2 + d (- 1) 3 a + b (1) + c (1) 2 + d (1) 3 a + b (3) + c (3) 2 + d (3) 3 . It is easy to see that A = 1- 3 9- 27 1- 1 1- 1 1 1 1 1 1 3 9 27 . 5.5 #4. Let the following matrix act on C 2 . Find the eigenvalues and a basis for each eigenspace in C 2 . 5- 2 1 3 . Solution. 5- - 2 1 3- = (5- )(3- ) + 2 = 2- 8 + 17 = 0 2 1 , 2 = 8 64- 4 * 17 2 = 8 2 i 2 = 4 i. For 1 = 4 + i we have 5- (4 + i )- 2 1 3- (4 + i ) a + bi c + di = ( a + b- 2 c ) + (- a + b- 2 d ) i ( a- c + d ) + ( b- c- d ) i = . This gives us four equations: a + b- 2 c = 0- a + b- 2 d = 0 a- c + d = 0 b- c- d = 0 We can now set up the matrix corresponding to the above system of linear equations 1 1- 2- 1 1- 2 1- 1 1 1- 1- 1 - 1 1- 2 2- 2- 2- 1 1 1 1- 1- 1...
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assign5-sol - Math 235 Assignment 5 Due 9:15am, Wednesday...

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