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Unformatted text preview: Lecture 10 Some Facts About Linear Systems Some inconvenient truths In the last lecture we learned how to solve a linear system using Matlab. Input the following: > A = ones(4,4) > b = randn(4,1) > x = A\b As you will find, there is no solution to the equation A x = b . This unfortunate circumstance is mostly the fault of the matrix, A , which is too simple, its columns (and rows) are all the same. Now try: > b = ones(4,1) > x = [ 1 0 0 0] > A*x So the system A x = b does have a solution. Still unfortunately, that is not the only solution. Try: > x = [ 0 1 0 0] > A*x We see that this x is also a solution. Next try: > x = [ -4 5 2.27 -2.27] > A*x This x is a solution! It is not hard to see that there are endless possibilities for solutions of this equation. Basic theory The most basic theoretical fact about linear systems is: Theorem 1 A linear system A x = b may have 0, 1, or infinitely many solutions....
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