110exam1solutions - Math 110 Exam 1 Name: Summer 2008 SID:...

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Unformatted text preview: Math 110 Exam 1 Name: Summer 2008 SID: Each part of each problem is worth the number of points stated in parentheses. A “medium” problem is denoted by an asterisk after the point total, and a “hard” problem is denoted by two asterisks after the point total. You must show all work to get any partial credit, which will be awarded for certain progress in a problem only if no substantially false statements have been written. If you are using a theorem, please write “By a theorem..,” and if you remember the name of the theorem, please write “By the (—–) Theorem...,” etc. If you are using a homework problem, please write “By a homework problem...,” and if you remember which homework problem, please write “By homework problem x.y.z ,” etc. You may not say that a certain problem follows from a theorem/homework problem if the theorem/homework problem is exactly what you are being asked to prove (for example, problems 3.1.1 and 3.2.1 on this exam). If you are asked to supply an example, you must prove your example is, in fact, and example. FOR INSTRUCTOR’S USE ONLY: Points Definitions 1 2 3 Theorems 1 2 3 Points Problems 1 a 1 b 2 a 2 b 3 a 3 b 3 c 4 a 4 b 1 1 Definitions Define the following terms/symbols. You may state results without proof in this section. Problem 1.1. ( 2 points ) Dimension of a vector space. Answer: Let V be a vector space. If there is a finite basis B of V , then we define dim( V ) = | B | . This is well defined as all bases of V have the same number of elements. If V does not have a finite basis, then we say V is infinite dimensional. Problem 1.2. ( 2 points *) [ T ] C B . Answer: Suppose V and W are finite dimensional vector spaces with ordered bases B = { v 1 ,...,v n } and C = { w 1 ,...,w m } respectively, and suppose T ∈ L ( V,W ). Then [ T ] C B ∈ M m × n ( F ) is the matrix given by [ Tv 1 ] C ··· [ Tv n ] C ....
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This note was uploaded on 03/18/2012 for the course MATH 110 taught by Professor Gurevitch during the Fall '08 term at Berkeley.

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110exam1solutions - Math 110 Exam 1 Name: Summer 2008 SID:...

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