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Lesson 9b -Homogenous Systems ^ Particular Solution

Lesson 9b -Homogenous Systems ^ Particular Solution -...

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23/09/2013 1 Homogenous System A Homogenous System is a linear system Ax=b, where b = 0 (the vector of all zeros). A homogenous system is always consistent! (It may be one solution, or infinitely many solutions, but never no solution). Why? Ax=0 will always have the solution x = 0 (A matrix multiplied by the zero vector will always produce the zero vector). Solution Sets for Homogenous Systems Theorem: Any homogenous system with k free variables will have the following solution set: } ,..., , | ... { 2 1 2 2 1 1 R t t t v t v t v t S k k k Where our t i are the free variables, and our v i are non zero vectors. Why? Well any free variable (x j ) we must substitute x j = t i (depending on the free variable location, we may have to skip some basic variables before getting to the next free variable). This means for each free variable, we introduce a new parameter, and the vector (v i ) will (at minimum) have the jth entry as a 1.
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23/09/2013 2 Example If we row reduce a matrix and it looks like the following: 0 0 0 0 1 0 0 0 0 2 1 0 2 0 1 Here we see that x 3 and x 4 are free. Which means we will assign them each a different free variable. In this case let us call them x 3 =t 1 and x 4 =t 2 . Right away we can see that the number of free variables will match the number of parameters we have. If we continue to solve for the basic variables we get: x 1 = 2t 1 and x 2 = 2t 1 t 2 . This gives us the solution set as follows:
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