EXAM IIMath 51, Spring 2003.You have 2 hours.No notes, no books, no calculators.YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONINGTO RECEIVE CREDITGood luck!NameID number1. (/30 points)2. (/30 points)3. (/30 points)4. (/30 points)5. (/30 points)Bonus (/15 points)Total(/150 points)“On my honor, I have neither given norreceived any aid on this examination. Ihave furthermore abided by all otheraspects of the honor code with respect tothis examination.”Signature:Circle your TA’s name:Byo ung-du Kim (2 and 6)Ted Hwa (3 and 7)Jacob Shapiro (4 and 8)Ryan Vinroot (A02)Michel Grueneberg (A03)Circle your section meeting time:11:00am 1:15pm 7pm11. (a) Find bases for the kernel and the image of the linear transformation given byTxyz=3x − y + zy + 2z3y + 6z2(b) Find any nonzero vector−→x with the property that−→x is perpendicular to everyvector in the kernel of T ; and explain how you know that the vector yo u supply hasthis property.32. (a) Suppose that A is a 4x3 matrix, and that C(A) has dimension 2. What is the di-mension of N (A)?(b) Suppose that N(M2) = {−→0 }. Show that N(M2M1) = N(M1).4(c) Suppose that M2is a 3x2 matrix, M1is a 2x4 matrix, t he rank of M2is 2, and therank of M2M1is 1. What is the dimension of C(M1)?Suggestions: Use part (b) and the Rank-Nullity Theorem to determine the dimensionsof N(M2) and N(M2M1), and then deduce the dim ension of N(M1).53. (a) Compute the determinant of the following matrix:A =3 −4 7 9 54 0 0 0 017 21 −5 11 60 0 2 0 00 3 0 0 0(Suggestion: Use the properties of the determinant to simplify this question beforeyou begin with the computations.)6(b) Is the above matrix A invertible? Why or why not?(c) Suppose B is a 2x2 matr ix with determinant 3, and that S is a set in R2such thatthe area of B(S) is 10.What is the area of S?74. Suppose that B is an invertible 3x3 matrix, but we are only given the first two columns:B =1 0 ?0 0 ?−4 2 ?(a) Is the above enough information to determine the first column of B−1? If so, findthat first column; if not, explain why you cannot find it.(Hint: What is B−110−4? What is B−1002?)(b) Is the above enough info rmation to determine the second column of B−1? If so, findthat second column; if not, explain why you cannot find it.85. For this problem, we define the following linear transformations in the plane:Rθ= rotation counter-clockwise aro und the origin by an angle θFθ= flip (reflection) over the line Lθobtained by rotating the x-axis counterclockwiseby the angle θF = F0= flip over the x-axis itselfv00vF(v)L0vF (v)0xyxyxy0R)v((a) Write down the matrices that correspond to Rθand F .9(b) Show that for any angle θ, we have Rθ◦ F = F ◦ R−θ(c) It can be shown also that Fθ= Rθ◦ F ◦ R−θ. Use this f act (you do not need to proveit) and the result from part (b) to answer the following:The composition Fα◦ Fβof two flips is the same as a rotation around the origin bywhat angle?10Bonus Question: Use linear algebra to help you find the area of the shaded region below,bounded by an ellipse centered at the origin, and two lines (make sure to b e explicit in howyo u are using linear algebra results!).37723 32( )232(
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