Midterm 1-Fall 2005

Midterm 1-Fall 2005 - AMS 510.01 Fall 2005, Midterm #1...

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Unformatted text preview: AMS 510.01 Fall 2005, Midterm #1 Name: Student ID: Score: /100 ( + /15 bonus) 1. Indicate whether each of the following statements is true or false. m and n should be taken as any constant positive integer. (2 points each). (a) ~u + ~v = ~v + ~u for all ~u,~v V , where V is any vector space. (b) The set of all square matrices forms a field. (c) ~u ~v = ~v ~u for all ~u,~v C n . (d) The set of all vectors ~x R n forms an algebra. (e) The expression f ( X ) = 2 X 2 + X + 5 is defined when X is any n n (square) matrix. (f) The set of all m n matrices forms a vector space. (g) If f ( a~u + b~v ) = af ( ~u ) + bf ( ~v ) for all a,b K and all ~u,~v V , then f ( ~x ) is a linear map acting on V. (h) If AB = I then A and B must be square matrices. (i) { (1 ,- 1) , (1 , 0) , (1 , 1) } is a basis for R 2 . (j) ( ~u ~v ) ~u = ( ~u ~v ) ~v = 1 for all ~u,~v R 3 ....
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Midterm 1-Fall 2005 - AMS 510.01 Fall 2005, Midterm #1...

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