Homework 2

This HW is due on 25th September 2024. Please write down the solutions neatly in your own handwriting, and submit the completed HW by email by the beginning of class on 25th September 2024. Please submit the HW as a single pdf file. To get credit, please provide all details and give complete reasoning for all your work.

Exercise 1 (3 points) For which three numbers k, does Gauss elimination break down? Which is fixed by a row exchange? In each case, is the number of solutions 0, 1, or \infty? \begin{aligned} kx_1 + 3x_2 &= 6\\ 3x_1 + kx_2 &= -6\\ \end{aligned}

Exercise 2 (2 points) Solve the following system by Gauss elimination (set up the augmented matrix and convert it to row echelon form, then do back substitution to find the solution). Circle the pivots. \begin{aligned} 2x_1 + 3x_2 + x_3 &= 8\\ 4x_1 + 7x_2 + 5x_3&= 20\\ -2x_2 + 2x_3 &= 0. \end{aligned}

Exercise 3 (5 points) A theorem we discussed in class (see LU factorization) say: “A matrix A (not necessarily square) is regular iff it can be factored as A=LU, where L is a lower triangular matrix with all diagonal entries 1 and U is an upper triangular matrix with all diagonal entries non-zero (these are the pivots of A). The non-zero off-diagonal entries in L (l_{ij} for i>j) are a record of the elementary row operations that transform A to U.”

Describe - in words - exactly what is the row operation corresponding to the non-zero off-diagonal entry l_{ij} in the i-th row and j-th column (i > j). Demonstrate your answer with a 3 \times 3 example.

Exercise 4 (5 points - optional) Modify the Gauss Elimination algorithm showed in class to incorporate partial pivoting. Add your own test case. Attach the .py or .ipynb file.