Chapters_5_and_6-Gaussian Elim & Mesh Analysis

Chapters_5_and_6-Gaussian Elim & Mesh Analysis -...

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Systems of Linear Equations: Gaussian Elimination For the sake of simplicity, we restrict ourselves to three or four unknowns. The basic ideas extend to arbitrary dimensions. Matrix Representation of a Linear System Matrices are helpful in rewriting a linear system in a very simple form. The algebraic properties of matrices may then be used to solve systems. First, consider the linear system 1
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Set the matrices 2
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Using matrix multiplications, we can rewrite the linear system above as the matrix equation The matrix A is called the coefficient matrix . The matrix C is called the right-hand-side vector . X is the vector of unknowns. The augmented matrix is the matrix [ A | C ], where 3
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ELEMENTARY OPERATIONS . It is clear that if we interchange two equations, the new system is still equivalent to the old one. If we multiply an equation with a nonzero number, we obtain a new system still equivalent to old one. 4
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And finally replacing one equation with the sum of two equations, we again obtain an equivalent system. These operations are called elementary operations Example. Consider the linear system The idea is to keep the first equation and work on the last two. In doing that, we will try to eliminate one of the 5
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unknowns and solve for the other two. For example, if we keep the first and second equation, and subtract the first one from the last one, we get the equivalent system Next we keep the first and the last equation, and we subtract the first from the second. We get the equivalent system 6
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Now we focus on the second and the third equation. We repeat the same procedure. Try to eliminate one of the two unknowns ( y or z ). Indeed, we keep the first and second equation, and we add the second to the third after multiplying it by 3. We get This obviously implies z = -2. From the second equation, we get y = -2, and finally from the first equation we get x = 4. Therefore the linear system has one solution 7
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Going from the last equation to the first while solving for the unknowns is called back substitution . Consider the augmented matrix corresponding to the original set of equations of the example. Let us perform some
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Chapters_5_and_6-Gaussian Elim & Mesh Analysis -...

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