Homework #1. Due January 15 |

Problems 1.* are from the Review on p.60, prblems 1.x.* are from section x in chapter 1.

Part A: Strang 1.4.15, 1.5.13, 1.1, 1.14, 1.19, 1.27, 1.29

Part B: Strang 1.3.11, 1.4.14, 1.4.17, 1.6.19, 1.6.23 and the following 2 problems:

6. Is it true that for a pair of elementary permutation and elementary lower triangular matrices L and P there exists another pair of such matrices L' and P' so that L' P' = P L ?

a) Check whether it is true in general.

b) Check whether it is true if

* L corresponds to an elimination step in the Gaussian method,

* P corresponds to a permutation step in the Gaussian method,

* the elimination step precedes the permutation step.

Note: we do not assume here that the permutation step follows IMMEDIATELY after the elimination step in the Gaussian method, there might be other steps between them.

7. a) Prove that a product of two lower triangular matrices L' and L'' is again a lower triangular matrix L' L'' = L

b) Prove that a product of two permutation matrices P' and P'' is again a permutation matrix P' P'' = P