Linear Algebra With Applications - 8 Edition - Chapter 4 - Problem 2
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# Set A = triu(ones(5)) tril(ones(5)). If L denotes the linear operator defined by L(x) =

Linear Algebra with Applications | 8th Edition

Problem 2

Set A = triu(ones(5)) tril(ones(5)). If L denotes the linear operator defined by L(x) = Ax for all x in Rn, then A is the matrix representing L with respect to the standard basis for R5. Construct a 5 5 matrix U by setting U = hankel(ones(5, 1), 1 : 5) Use the MATLAB function rank to verify that the column vectors of U are linearly independent. Thus, E = {u1, u2, u3, u4, u5} is an ordered basis for R5. The matrix U is the transition matrix from E to the standard basis. (a) Use MATLAB to compute the matrix B representing L with respect to E. (The matrix B should be computed in terms of A, U, and U1.) (b) Generate another matrix by setting V = toeplitz([1, 0, 1, 1, 1]) Use MATLAB to check that V is nonsingular. It follows that the column vectors of V are linearly independent and hence form an ordered basis F for R5. Use MATLAB to compute the matrix C, which represents L with respect to F. (The matrix C should be computed in terms of A, V, and V1.) (c) The matrices B and C from parts (a) and (b) should be similar. Why? Explain. Use MATLAB to compute the transition matrix S from F to E. Compute the matrix C in terms of B, S, and S1. Compare your result with the result from part (b).

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1/17/18 Lecture Parametric Equations It is difficult to represent an arbitrary curve with traditional functions y = f(x) or x = g(y). We create a new method to represent such a curve, by using parametric equations. We represent x and y in terms of a third variable (t, theta, etc.). We must also specify bounds of our parameter, in this case t. Our bounds can be positive and negative infinity (representing no bounds on t). Bounds can be used to restrict a curve to end at specific points. Graphing parametric equations Method 1: Table. - Create a table of values for t, x, and y. T X Y ­1 3 5 0 1 4 1 ­1 5 2 ­3 9 We can then plot the points (x,y) on our graph and sketch our curve. It has a direction of motion, a positive direction that r

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