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Unformatted text preview: Math 307, Section 102 8 Problems for a total of 85 points Final Exam Friday, December 15, 2006. No books, notes or calculators Problem 1. (10 points) (The scalar field is R .) Consider the matrix A and column vector ~ b : A = 1 1 1 0 0 1 2 3 4 ~ b = 1 2 3 (a) (6 points) Find the PA = LU factorization for A . (b) (2 points) Recall that the PA = LU factorization can be used to rewrite the system A~x = ~ b as two systems with triangular coefficient matrix. Write down these two triangular systems. (c) (2 points) Solve the two triangular systems to find all ~x such that A~x = ~ b . Problem 2. (10 points) (The scalar field is R .) Consider the matrix A = 1 2 3 4 2 4 3 0 5 1 2- 1 5 5 (a) Find a basis for the nullspace of A . (b) Find a basis for the column space of A . (c) Find a basis for the left nullspace of A . (d) Find a basis for the row space of A ....
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