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1.2 Homework Solutions - Math 307: Problems for section 1.2...

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Math 307: Problems for section 1.2February 2, 2009Many problems in this homework make use of a few MATLAB/Octave.mfiles that are provided onthe website. In order to use them, make sure that the files are in the same directory that you are runningMATLAB/Octave from (to see which directory this is, typepwdin MATLAB/Octave).1.Compute the determinant of a4×4Vandermonde matrix. Bonus: show that the generalformula for the determinant of a Vandermonde matrix is correct.Here is the general calculation, although you only need to hand in the 4×4 version of this.Themain point is that you can write the determinant for then×nmatrix as an expression involving thedeterminant of the (n-1)×(n-1) matrix. If you start with the 4×4 case, you can then substitutethe expression we already computed for the 3×3.The general case goes by induction. We know the formula is true forn= 1 (alson= 2,3). The inductivestep is to show that the formula fornfollows from the formula forn-1. Letd(n;x1, x2, . . . , xn) be thedeterminant of then×nVandermonde matrix with variablesx1, x2, . . . , xn. We begin with the samesteps as the 3×3 example done in lectures. First we subtractxntimes the second column from thefirst column, thenxntimes the third column from the second column, and so on. This doesn’t changethe determinantd(n;x1, x2, . . . , xn) = detxn-11xn-21· · ·x21x11xn-12xn-22· · ·x22x21xn-13xn-23· · ·x23x31..................xn-1nxn-2n· · ·x2nxn1= detxn-11-xn-21xnxn-21-xn-31xn· · ·x21-x1xnx1-xn1xn-12-xn-22xnxn-22-xn-32xn· · ·x22-x2xnx2-xn1xn-13-xn-23xnxn-23-xn-33xn· · ·x23-x3xnx3-xn1.

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Term
Fall
Professor
RICHARDFROESE

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