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Midterm 2-Fall 2007

# Midterm 2-Fall 2007 - AMS 510.01 Fall 2007 Midterm#2 Name...

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AMS 510.01 Fall 2007, Midterm #2 Name: Student ID: Score: /100 ( + / 10 bonus) 1. Indicate whether each of the following statements is true or false. (2 points each). a) Every real matrix, A , can have an eigenvalue decomposition (given by = where D is diagonal). b) If we exchange rows 1 and 2 of A, the determinant of A will stay the same. c) Consider the transformation : × × defined by ( ) = (where A is an arbitrary × matrix). F can be represented by a matrix, B , such that ( ) = . d) An × real symmetric matrix always has m distinct real eigenvalues. e) lim = 1 . f) If lim = , then we can always find a number, N, such that | | > | | for all > . g) sgn ({2, 4, 3, 1})= 0. h) If f is continuous at , then lim ( ) = (lim ) . i) det 2 2 1 1 2 0 3 0 1 = −3 . j) Let : → and : → be any linear mappings, then ( + ) = ( ) + ( ).

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2. Consider the linear map : defined by: ( , , ) = ( + 2 , + 2 , + 2 ) a) Find the dimension and a basis for the kernel and image of F . (10 points) b) Decide if F has an inverse mapping. If so, give the inverse mapping; otherwise, give the reason. (5 points)
3. Consider the linear transformation, : , defined by:

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Midterm 2-Fall 2007 - AMS 510.01 Fall 2007 Midterm#2 Name...

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