Sample07I - Prof. Ming Gu, 861 Evans, tel: 2-3145 Office...

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Unformatted text preview: Prof. Ming Gu, 861 Evans, tel: 2-3145 Office Hours: MWF 3:00-4:00PM Email: mgu@math.berkeley.edu http://www.math.berkeley.edu/∼mgu/MA54 Math54 Sample Midterm I, Fall 2007 This is a closed book, closed notes exam. You need to justify every one of your answers unless you are asked not to do so. Completely correct answers given without justification will receive little credit. Look over the whole exam to find problems that you can do quickly. You need not simplify your answers unless you are specifically asked to do so. Hand in this exam before you leave. Problem Maximum Score 1 2 19 6 19 Total Your SID: 19 5 Your GSI: 19 4 Your Name: 19 3 1. (5 Points) 5 100 Your Score Math54 Sample Midterm I, Fall 2007 2 2. (19 Points) (a) Solve linear systems of equations A x = b, where 123 A= 1 2 4 211 6 and b = 7 . 4 (b) Consider linear systems of equations A x = b, where 124 A = 1 2 k2 211 6 and b = 3k . 4 For what values of k does the system have a unique solution? infinite number of solutions? no solution? Math54 Sample Midterm I, Fall 2007 3 3. (19 Points) Let P be the set of all functions of the form c0 + c1 sin(x) cos(x) + c2 cos2 (x) + c3 sin2 (x), where the c’s are arbitrary real constants. It is known that P is a linear space under the usual function addition and scalar multiplication. Find the dimension and a basis for P . Math54 Sample Midterm I, Fall 2007 4 4. (19 Points) Let u1 , · · · , um be vectors in span{v1 , · · · , vk }; and let v1 , · · · , vk be vectors in span{w1 , · · · , wn }. Show that u1 , · · · , um are vectors in span{w1 , · · · , wn }. Math54 Sample Midterm I, Fall 2007 5. (19 Points) If the image of an n × n matrix A is Rn , show that A must be invertible. 5 Math54 Sample Midterm I, Fall 2007 6 6. (19 Points) Find examples of n × n matrices A and B such that A, B are not invertible but A + B is. ...
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Sample07I - Prof. Ming Gu, 861 Evans, tel: 2-3145 Office...

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