review3 - MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2010,...

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Warning : This is only a partial review. You still need to read the textbook to see all the topics we have covered. Read key to HWs and do more practices using exercises on the textbook. 1. How to use the Gauss elimination process to get the reduced echelon form ? Practice Problem : Let A = 1 - 2 - 1 1 2 - 3 1 6 3 - 5 0 7 1 0 5 9 . (a) Find the row-reduced echelon matrix R of A . (b) Find the elementary matrices E 1 ,...,E k such that E k ··· E 1 A = R. 2. Solving homogeneous system of linear equations Ax = 0, where A is a m × n matrix: Its solution space is a vector space , which is the same as the nullspace of L A . The dimension of the solution space is n - rank ( A ). Solutions x (complete solutions)of Ax = 0 can be written as a linear combinations of the vectors in a basis of the solution space, i.e. x = c 1 x 1 + ··· + c k x k , where { x 1 ,...,x k } is a basis for the solution space, and k = n - rank ( A ). Ax = 0 is equivalent to Rx = 0 where R is the educed echelon form of A . Hence a basis for the solution space { x 1 ,...,x k } can be found by looking at R . Practice problem (see HW#9) : Suppose A has row reduced form R , A = 1 2 1 b 2 a 1 8 ? ? ? ? , R
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review3 - MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2010,...

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