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worksheet11

worksheet11 - Review and Examples(Spr 2006 Due Thursday 6th...

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Review and Examples (Spr. 2006) Due Thursday 6th April These are examples of questions with the degree of difficulty that you will encounter in the lab-exam. Make sure that you can do them! Use Mathematica to solve all parts of the problems Vectors Problem 1. Show that for any vectors ~a , ~ b , ~ c that, (i) ~a. ( ~a ~ b ) = 0 (ii) ~a. ( ~ b ~ c ) = ( ~a ~ b ) .~ c Lists and Matrices Problem 2. Display the following matrix in matrixform. matrix = {{ 0 . 2 , - 0 . 4 , 0 . 1 } , { 0 . 4 , 0 . 2 , - 0 . 3 } , { 0 . 6 , - 0 . 3 , - 0 . 1 }} We haven’t found the eigenvalues and vectors of symmetric matrices be- fore; however, this illustrates the power of mathematica. Look up Eigenvalue in the help manual. You won’t fully appreciate the power of this until you have to solve eigenvalue problems by hand in advanced mechanics and quan- tum mechanics. Find the determinant, eigenvalues and eigenvectors of the matrix matrix . Show that the sum of the eigenvalues (= the trace of the matrix) and the product of the eigenvalues (= the product of the eigenvalues) are both real even though the eigenvalues themselves are complex. Function definitions f [ x ] Problem 3. Define the function f ( x ) = 0 . 8 x + x 2 - 3 . 2 x

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worksheet11 - Review and Examples(Spr 2006 Due Thursday 6th...

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