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Unformatted text preview: FINAL EXAM PRACTICE I, May 8, 2010 Math 21b, Spring 10 Name: MWF10 Ryan Reich MWF10 Stewart Wilcox MWF11 Oliver knill MWF11 Ryan Reich MWF11 Stewart Wilcox MWF12 Juliana Belding TTH10 Wushi Goldring TTH1130 Alex Subotic Please write your name above and check your sec- tion to the left. Try to answer each question on the same page as the question is asked. If needed, use the back or the next empty page for work. If you need additional paper, write your name on it. Do not detach pages from this exam packet or un- staple the packet. Please write neatly and except for problems 1-3, give details and justifications. Answers which are illegible for the grader can not be given credit. No notes, books, calculators, computers, or other electronic aids can be allowed. You have 180 minutes time to complete your work. 1 20 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 Total: 140 Problem 1) (20 points) True or False? No justifications are needed. 1) T F If A is a real n n matrix, then A + A T is diagonalizable. 2) T F The equation z 3 = 1 has three different complex solutions z and z = 1 is one of them. 3) T F There is a 2 2 projection matrix A that projects onto a line and a 2 2 reflection matrix B that rotates about a point, so that AB is a rotation matrix. 4) T F The transformation T ( f )( x ) = f ( f ( x )) is linear on the space C of all smooth functions 5) T F If a continuous function f defined on the interval [ , ] is both even and odd, then it must be constant function. 6) T F If A is a 2 2 matrix, then the characteristic polynomial of AA T is 2 + c for some real constant c . 7) T F The function f ( t ) = e t is an eigenfunction with eigenvalue 1 of the linear operator T = D 2 where Df = f . 8) T F The initial value problem f ( x ) + f ( x ) = sin( x ) ,f (0) = 1 ,f (0) = 1 has exactly one solution. 9) T F The transformation L ( A ) = A + A T is a linear transformation on M n and L has only the eigenvalues 0 and 2. 10) T F The set X of smooth functions f ( x,y,z ) which satisfy the f xx + f yy + f zz = f is a linear space....
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