Last Name First Name MATH 23a FALL 2003 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Final Exam January 24 2004 Directions You have three hours for this exam though I have designed it to take less than the full amount of time No calculators notes books etc are allowed Please answer on the pages provided there are blank pages included for scratch work Note that not all questions and not all parts are equally weighted Problem Points Score 1 44 2 16 3 12 4 10 5 20 6 20 7 0 Total 122 Please note that question 7 is for your amusement only in case you finish with time to spare 1 True or False 44 points 2 each T or F There exist non trivial vector spaces with only finitely many vectors T or F If L Rn Rm is a linear transformation and n m then Ker L is non trivial T or F If A U V and B V W are both injective linear maps then B A U W is also injective T or F If U and V are subspaces of a vector space then U V V U U V T or F Every skew symmetric multilinear form f V n F is alternating T or F If dim V m and f V n F is an alternating form with n m then f 0 T or F If A On R then det A 1 T or F Every Cauchy sequence of integers converges to an integer T or F Every bounded set of integers has a least element T or F Every bounded set of rational numbers has a least element T or F Every bounded set of real numbers has a least element T or F If A B Mn R then det AB det BA T or F If A B Mn R are similar then det A det B T or F If A B Mn R are similar then Spec A Spec B T or F If 123 456 S7 then sgn 1 T or F If 123 456 S7 then is bijective as a function from the set X 1 2 3 4 5 6 7 to itself T or F If 123 456 S7 then 1 T or F Given a linear transformation A V V any set of non zero eigenvectors for A with distinct eigenvalues is linearly independent T or F Given an inner product space V any set of mutually orthogonal non zero vectors is linearly independent T or F If V is a subspace of an inner product space U then V V 0 T or F If V is an inner product space and u v V then u v u v T or F If V is an inner product space and u v V satisfy hu vi 0 then either u 0 or v 0 2 Completeness 16 points 4 4 8 a State the Completeness Axiom for the Real Numbers b Give an example of a Cauchy sequence of rational numbers that does not converge to a rational number c Let ai bi R be a closed interval for each i N Such a collection of closed intervals is said to be nested if ai 1 bi 1 ai bi for each i N Given a collection of nested closed intervals as above let I ai bi i 1 be the intersection of these intervals Show that I 6 3 Cross Products 12 points 4 4 4 0 For u a b c v x y z R3 we define the cross product of these vectors to be u v bz cy cx az ay bx a Show that the cross product is a bilinear map from R3 R3 R3 b Show that the cross product is skew symmetric c Show that if R3 is given the usual inner product the dot product then u v is orthogonal to u d Bonus Fact 0 points It is true but you don t have to prove that u v u v sin where is the angle between the two vectors in the plane spanned by u and v 4 A Symmetric Matrix 10 points 5 each x 1 0 Let A 1 x 1 M3 R 0 1 x a For what real values of x is A invertible Explain b For what real values of x is A orthogonal Explain 5 A Jordan Block 20 points 10 5 5 We have seen in class that some but not all matrices are diagonalizable For a matrix that is not one of the next best results along these lines would be to be able to put such a matrix in Jordan Canonical Form In this problem we tackle one of the building blocks for this result Let V R2 and let A V V Suppose that the characteristic polynomial of A is pA 2 a Show that exactly one of the following possibilities must hold A is diagonalizable What is the diagonalized form of A There is a basis for V with respect to which 1 A 0 In the case when 1 the linear transformation A is known as a shear Of course the characteristic polynomial implies that 1 is the only eigenvalue for a shear For the following let B be a shear b Use part a and a change of basis to show that B I 2 0 Note that B does not already have the form in part a c Use part b to show that Bx x V1 for every x R2 where V1 is the eigenspace corresponding to the eigenvalue 1 6 Nilpotent Matrices 20 points 5 each Let V Rn and let A V V We say that A is nilpotent if there is some m N such that Am 0 and we say that m is the degree of nilpotency if Am 0 but A m 1 6 0 a Let A V V be nilpotent of degree m Let v V be such that A m 1 v 6 0 Show that v Av A2 v A m 1 v is a linearly independent set in V Conclude that m n b Exhibit examples of nilpotent matrices of degrees 1 2 and 3 c Let A V V be nilpotent Find all the eigenvalues of A d Show that if A is nilpotent then I A is invertible If A is nilpotent then I A is called unipotent 7 Un digestif Invertible Matrices over Finite Fields 0 points This is just something to think about in case you have already finished the rest of the exam It is not extra credit because I will not be grading it no matter what you write Recall that if Mn F is the collection of n n matrices with entries from the field F then we define GLn F A Mn F det F 6 0 If F Z pZ where p is prime then what is GLn F for n 1 2 and 3 Recall that X is the cardinality of the set X Hint Consider the column vectors of an …