HW_6_SolutionGuide

HW_6_SolutionGuide - ECE280 Fall 2009 F Salem HW#6Solution...

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ECE280 Fall 2009 F. Salem HW#6—Solution Guide—using diary on command in MATLAB %Given X1=[1 2 3]' X1 = 1 2 3 X2=[4 5 6]' X2 = 4 5 6 X3=[7 8 9]' X3 = 7 8 9 X4=[7 8 10]' X4 = 7 8 10

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% and given the matrices B=[X1 X2 X4] B = 1 4 7 2 5 8 3 6 10 L=diag([11, 12, 13]) L = 11 0 0 0 12 0 0 0 13 A=B*L*inv(B) A = 19.6667 -13.3333 6.0000 9.3333 -2.6667 6.0000 12.0000 -18.0000 19.0000 %Problem 1, solution. Check if X1, X2, X3 are linearly %independent. %To do so, we form the matrix, say, F=[X1 X2 X3] and check %if its rank equals to 3 (the number of vectors). Only if %the rank equals to the number of vectors would the set of %vectors X1, X2, and X3 be linearly independent. F=[X1 X2 X3] F = 1 4 7 2 5 8 3 6 9 rank(F) ans = 2 %rank (F)=2 less than 3 (the number of vectors). Then, the %vectors X1, X2, and X3 are NOT linearly independent.
%Problem 2- solution: Select 3 vectors and check the rank %of matrix formed from them. %When the rank of the matrix equals to 3 the set of 3 vectors are linearly independent. %Thus, we proceed as follows: F1=[X1 X2 X4] F1 = 1 4 7 2 5 8 3 6 10 rank(F1) ans = 3 % Therefore, the vectors X1, X2, and X4 are linearly %independent. You may get other set of vectors. % It is OK as long as they are linearly independent. %We call these 3 vectors a BASIS since they are linearly %independent and span the 3-dimensional space. % %Problem 3-solution. The vectors X=F1*X_bar, where X_bar is %the new representation. % since rank(F1)=3 (full rank) and F1 is a square matrix, %then F1 is invertible. %Therefore, we can easily solve for X_bar as %X_bar=(inv(F1))*X. First, write X in the natural %representation. X=[1 0 0;0 1 0;0 0 1]*[7 8 10]' X = 7 8 10

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X_bar=inv(F1)*X X_bar = 0.0000 0 1.0000 %4. eigen-values/vectors of A. (i) Solve det(Lambda*I-A)=0, %for the eigenvalues, then (ii) for each eigenvalue solve %for the (non-zero) vector that satisfies %[Lambda_i*I-A]*V_i=0 (This gives 3 equation in 3 unknown
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This note was uploaded on 11/04/2009 for the course ECE 280 taught by Professor Mukkamala/udpa during the Spring '08 term at Michigan State University.

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HW_6_SolutionGuide - ECE280 Fall 2009 F Salem HW#6Solution...

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