Prof. Ming Gu, 861 Evans, tel: 2-3145Office Hours: MWF 3:00-4:00PMEmail: [email protected]://www.math.berkeley.edu/∼mgu/MA54Math54 Sample Midterm I, Fall 2007This is a closed book, closed notes exam. You need to justify every one of youranswers unless you are asked not to do so. Completely correct answers given withoutjustification will receive little credit. Look over the whole exam to find problems thatyou can do quickly. You need not simplify your answers unless you are specificallyasked to do so. Hand in this exam before you leave.Problem Maximum Score Your Score1 52 193 194 195 196 19Total 1001. (5 Points)Your Name:Your GSI:Your SID:Math54 Sample Midterm I, Fall 2007 22. (19 Points)(a) Solve linear systems of equations A x = b, whereA =1 2 31 2 42 1 1and b =674.(b) Consider linear systems of equations A x = b, whereA =1 2 41 2 k22 1 1and b =63k4.For what values of k does the system have a unique solution? infinite number ofsolutions? no solution?Math54 Sample Midterm I, Fall 2007 33. (19 Points) Let P be the set of all functions of the form c0+ c1sin(x) cos(x) + c2cos2(x) +c3sin2(x), where the c’s are arbitrary real constants. It is known that P is a linear spaceunder the usual function addition and scalar multiplication. Find the dimension and a basisfor P.Math54 Sample Midterm I, Fall 2007 44. (19 Points) Let u1, · · · , umbe vectors in span{v1, · · · , vk}; and let v1, · · · , vkbe vectors inspan{w1, · · · , wn}. Show that u1, · · · , umare vectors in span{w1, · · · , wn}.Math54 Sample Midterm I, Fall 2007 55. (19 Points) If the image of an n × n matrix A is Rn, show that A must be invertible.Math54 Sample Midterm I, Fall 2007 66. (19 Points) Find examples of n × n matrices A and B such that A, B are not invertible butA + B
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