final_solutions - EE263 Summer 2010-11 Laurent Lessard...

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Unformatted text preview: EE263 Summer 2010-11 Laurent Lessard EE263 final exam If you received this document via email, print out this page, sign it, and submit it with your completed final exam. You may scan and email your final if you like, but we prefer that you turn it in to the dropbox in Packard if possible. Instructions • You will need access to a computer with Matlab. • You may use the lecture slides, any notes you took yourself, and any material posted on the course website (including homework solutions). • You may also use supplementary material such as textbooks, wikipedia (at your own risk!), etc. but there shouldn’t be any need to consult such outside sources. • The exam is graded out of 70 points. Point values for problems are indicated at the start of each problem. Sub-parts are labeled (a), (b), etc. and are worth 3 points each for the first problem, and 5 points each for every other problem. • Turn in all your work, and be sure to attempt every problem. Submit all Matlab code that you write. • Be neat, and be concise. We will take points off if we can’t read your solutions or if they are unneces- sarily messy and/or verbose. • There are no hints available, so don’t ask for any. If you need clarification on one of the problems, please send your questions to: [email protected] • You must turn in your completed final exam to the grader no later than 24 hours after you picked it up or received it via email. • Some of the problems require you to download supplementary Matlab files. You can find all files related to the final exam here: • You are not permitted to work in groups, or discuss exam problems with anybody. You must work alone and turn in your own work. By signing below, you are asserting that you have read and understood the Stanford Honor Code, and promise to uphold it. Signature Date/Time received Name (printed) Date/Time submitted 1. Short Problems . [15 points] Suppose A ∈ R m × n , B ∈ R n × m , and Q ∈ R n × n . Provide a short proof for each of the following problems. (a) Show that ( I − AB )- 1 = I + A ( I − BA )- 1 B . You may assume that both inverses exist. Solution. Expand the product: ( I − AB ) bracketleftbig I + A ( I − BA )- 1 B bracketrightbig = I − AB + ( I − AB ) A ( I − BA )- 1 B = I − AB + ( A − ABA )( I − BA )- 1 B = I − AB + A ( I − BA )( I − BA )- 1 B = I − AB + AB = 0 Therefore ( I − AB )- 1 = I + A ( I − BA )- 1 B . (b) Suppose Rank ( A ) = 1. Show that A = uv T for some u ∈ R m and v ∈ R n . Solution. The rank is defined as the dimension of the range. So if range( A ) has dimension 1, that means that every column of A is a multiple of a 1 , the first column of A . In particular: a 2 = v 2 a 1 , a 3 = v 3 a 1 , ... , a n = v n a 1 . Therefore: A = bracketleftbig a 1 a 2 ... a n bracketrightbig = bracketleftbig a 1 v 2 a 1 ... v n a 1 bracketrightbig = a 1 bracketleftbig 1 v 2 ... v n bracketrightbig...
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final_solutions - EE263 Summer 2010-11 Laurent Lessard...

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