Unformatted text preview: Homework 5 (Due Wednesday, November 10)
Consider the following system: (i) Perform Gaussian / Gauss – Jordan Elimination on the augmented matrix. (ii) Why does this system have infinite solutions? (i. e. Explain in short the result of elimination.) (iii) Write the pivot variables and free variables. (iv) Write the solutions in parametric form. Your answer should contain 2 different parameters, e. g. s and t. (i.e. Write the pivot variables in terms of free variables.) 2. Exercises 3.4, Problem 13 (Hint: part (a): Exchange 2nd and 3rd rows in matrix A (and b), then convert the system into Leontief like system.) In part (b), carry out only 3 iterations. Exercises 3.5, Problem 7 (Refer to example 1, section 3.5. Do NOT perform elimination on 20-by-20 matrix. Try to observe the pattern in Gaussian Elimination of 6-by-6 Frog Markov chain. Then use the same pattern for 20-by-20 matrix.) Let be any vector in x – y plane. Find the 2-by-2 matrix A for each of the following Linear 3. 4. Transformations. (i) Project vector u on y – axis. (ii) Reflect vector u about y – axis. (iii) Project vector u on the line y = x. (iv) Rotate vector u anticlockwise through 30o, then project onto y – axis, and then stretch 2-times. 5. 6. Exercises 4.2, Problem 7 (Plot is not required) Consider the following set of data, which are believed to obey exponential law x y 1 14 2 49 3 171 4 600 (approximately). Perform a transformation y’=f(y) on y so that the regression model y’ = qx + r’ is fairly accurate. Determine q and r’, then reverse the transformation to determine r. ...
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- Spring '08
- Linear Algebra, Markov chain, Complex number, pivot variables