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HW-8-Solutions

HW-8-Solutions - ECE-S511 Systems I Fall 2012-2013...

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ECE-S511: Systems I Fall 2012-2013 7 December 2012 1 Dept. of ECE, Drexel University Assignment 8 Solutions Written assignment: 1. Solution is a linear operator and hence there exists a matrix that is naturally associated with it. We will assume this representation is w.r.t. the natural basis: { } { } . Thus we can write: ([ ]) [ ] [ ] [ ] [ ] Eqn [1] Here, subscript n represents the vector/matrix w.r.t the natural basis. We want however to find the representation of with the input and output given w.r.t the basis . Note that we know how to get the relationship: [ ] [ ] . is a matrix such that the column is the representation of the basis vectors in with respect to the basis set .
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ECE-S511: Systems I Fall 2012-2013 7 December 2012 2 Dept. of ECE, Drexel University Thus, representing the first and second column of w.r.t. the natural basis gives very easily the matrix as: [ ] We wish to have the following expression: ([ ]) [ ] , We can rewrite Eqn [1] above using the fact that transforms vectors from basis representation in to those in to get: [ ] [ ] [ ] ([ ]) If we multiply on the left by we get: [ ] ([ ]) This implies: [ ] To check if we are correct: I chose a random vector .
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