Midterm 2-Fall 2006

# Midterm 2-Fall 2006 - AMS 510.01 Fall 2006, Midterm #2...

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Unformatted text preview: AMS 510.01 Fall 2006, 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) The following matrix, A , is invertible: A = 4 2 1 3 7 2 3 2 1 1 2 (b) 2 1 1 1 2 3 1 =- 1 (c) F G = G F for any two linear maps F and G . (d) Every n-square real matrix has n real eigenvalues. (e) A matrix is diagonalizable if and only if it is non-singular. (f) &lt; ~u, ~v &gt; = &lt; ~v, ~u &gt; for all vectors ~u and ~v from any real inner product space. (g) The following matrix, B , is diagonalizable: B = 2- 4 7- 4 1 5 7 5 3 (h) An onto mapping can be defined from the set S = { 1 , 2 , 3 } to the set T = { 2 , 4 , 6 , 8 } . (i) For any non-zero vectors ~u and ~v , Proj( ~u, ~v ) = ~ 0 if and only if ~u and ~v are orthogonal. (j) sgn( { 2 , 3 , 1 } ) = 1. 2. Consider the linear map F : R 3 R 4 defined by: F ( x, y, z ) = ( x + y, x + z, z- x, z- y ); Find the dimension and a basis for the kernel and image of F . (10 points) 3. Consider the function3....
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## This note was uploaded on 02/09/2010 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.

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Midterm 2-Fall 2006 - AMS 510.01 Fall 2006, Midterm #2...

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