110mockfinal.pdf - Math 110 u201cPractice...

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Math 110 “Practice Final” Alex Youcis Instructions and disclaimer: The scare quotes in the title of this note are meant to scare away people who take this as being a practice final too literally. Namely, note that this note was not written with a distinct time limit in mind (so it may not be exam length) nor do I know better than you what will be on the final. Take this as a smattering of material that may or may not be on the final, no more. Again, let me belabor the point: this is not a strict indication of what will be or will not be on the exam. For example, it is likely more ‘theoretical’ than the final will be (again, just a guess) and, in particular, leaves o some important computational topics (e.g. Gram-Schmidt and finding Jordan bases). If one chooses to take this like an exam, one should note that for each problem there are “O1)”, “O2)”, etc. which are “option 1”, “option 2”, etc. They are generally problems of a similar nature (maybe some more theoretical and one more computational). One should, if one takes it as a practice exam, pick and choose from “O1)”, “O2)”, etc. on each problem allowing one to essentially make multiple practice finals (although, beware, some of the optioned problems are significantly easier if you have done some of the earlier options—so I suggest you do them sequentially). Finally, the number of T or F questions is exorbitant (for your benefit) and does not reflect the amount of T or F questions on the final exam. One should note that there is a non-trivial chance that there are errors (typographical or otherwise) in this note. So, read carefully and don’t take anything for granted. Alert me to any errors you might find. Lastly, some notational concerns: The notation ( - ) > means “the transpose of ( - )”. The notation M n ( F ) means the F -space of n n -matrices over F , thought of as operators on F n by their action on the standard basis of F n . Similarly, any n n matrix will be thought of as an operator C n unless stated otherwise. Any mention of an inner product space without specification is taken to be F n , where F is either R or C depending on context, with the standard inner product (i.e. h x, y i := x > y ) unless stated otherwise. The phrase ‘unitary’ below means ‘isometry’ in the parlance of our book (i.e. an operator A such that h Ax, Ay i = h x, y i for all x, y ). For an operator A , we denote by m A ( x ) the minimal polynomial of A , and by χ A ( x ) its characteristic polynomial. Two matrices A and B in M n ( F ) are similar , written A B , if there exists some invertible matrix S such that A = SBS - 1 . Equialently (check this as an exercise!) there exists an ordered basis B of F n such that [ A ] B = B . For an operator A on F and an element λ 2 F we denote by m g ( A, λ ) the geometric multiplicity of λ (in A ) defined to be dim F Null( A - λ I ). We denote by m a ( A, λ ) the algebraic multiplicity of λ (in A ) defined to be the largest power of x - λ that divides χ A ( x ).

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