Prof. Ming Gu, 861 Evans, tel: 2-3145Office Hours: TuTh 12:30-2:00PMEmail: [email protected]://www.math.berkeley.edu/∼mgu/MA54MMath54 Sample Midterm II, Fall 2007This is a closed everything exam, except a standard one-page cheat sheet (on one-side only). You need to justify every one of your answers. Completely correct answersgiven without justification will receive little credit. Problems are not necessarily orderedaccording to difficulties. You need not simplify your answers unless you are specificallyasked to do so.Problem Maximum Score Your Score1 52 193 194 195 196 19Total 1001. (5 Points) Write your p ersonal information below.Your Name:Your GSI:Your SID:Math54 Sample Midterm II, Fall 2007 22. Consider the following map from R2to R2:T xy!= x + y − 1x − y + 1!.Is T a linear transform? Explain.Math54 Sample Midterm II, Fall 2007 33. Is the following identity true for all pairs of invertible matrices A, B ∈ Rn×n?(A B)−1T=A−1TB−1T.Explain your answer.Math54 Sample Midterm II, Fall 2007 44. Give an explicit formula for the components of the vector −3 44 3!k 10!for any integer k > 0.Math54 Sample Midterm II, Fall 2007 55. Show that the following functionx · y = 3x1y1+ 2x2y2is an inner product on R2.Math54 Sample Midterm II, Fall 2007 66. LetV = span1111,1−1−11.Find an orthonormal basis for the orthogonal complement of V
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