MATH 307 Midterm: MATH 307 2006 Winter Test 1 sec101

Consider the matrix A = [2 1 -3 6 4 0 -4 2 1]. Calculate the LU-factorization of A. Use the LU-factorization you found in (a) to solve Ax = b for times in the following two cases: b = [0 10 -1], b = [1 12 -7] What is det(A)?

??-2 Assignment 3 0 3-4 51 8. For the matrix A-2 22 02 18 4 -6 a) Find a basis for the row space of A b) Find a basis for the column space of A 1 What is the rank of A? d) Find the nullspace of A and give the nullity of A

3 -1 4. Let A- 4 2 and b 20 10 a) Use the Gram-Schmidt process to find an orthonormal basis for the column space of A. (6 points) b) Factor A into a product QR, where Q has an orthonormal set of column vectors and R is upper triangular. (4 points) c) Solve the least squares solution of Ax b. (6 points)