Linear Algebra MAS 3105
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This 1 page Class Notes was uploaded by Wilburn Hayes on Monday October 12, 2015. The Class Notes belongs to MAS 3105 at Florida International University taught by Steven Hudson in Fall. Since its upload, it has received 11 views. For similar materials see /class/221786/mas-3105-florida-international-university in Math at Florida International University.
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Date Created: 10/12/15
Lecture Notes in MAS 3105 Thursday 24 2003 See Leon Ch 14 After the MATLAB session we went over a few final ideas about elementary matrices Most of this is in the text except maybe part 3 l Triangular factorization read pages 74 75 especially the concluding paragraphs The main idea is given a matrix A try to find triangular matrices L and U so that A LU How Do GE on A using only type III operations using lower triangular E7s until U is upper triangular if this is possible Since products and inverses of lower triangulars are still lower triangular proof omitted we get the LU factorization where L is lower triangular We did not go into detail on why this is useful but it helps in solving systems ef ciently and in designing circuits 2 Read over the TFAE theorem 143 and its proof We discussed the proof of each implication arrow of the 77logic triangle 1 gt b a c a a The first and third arrows are pretty easy see the text The idea of the second one is this Reduce Ax 0 to RREF U16 0 so A and U are row equivalent lf b is true and the solution is unique then there can be no free variables So there have to be 71 leading ones in the square nxn matrix U The only way this can happen is if U I which proves a This completes the proof of the arrows in the 77triangle77 and the proof of thml43 3 GE works because it replaces a system by an equivalent system it doesn7t change the solution set How do we know this Suppose the original system is Ax b GE is the same as multiplying both sides of the system by a lot of E7s or by a single matrix M the product of the E7s The new system is MAx Mb Clearly any solution of the first system is a solution of the new one Conversely if x solves the new system we can multiply that by M 1 which exists since the E7s are all nonsingular to see it solves the first system So the systems are equivalent End of lectures on Ghl On Tuesday 12903 we started Gh2 Determinants
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