topic3 - Systems of Linear Equations 1. Solve a x i =1 n ii...

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Systems of Linear Equations 1. Solve n i i i 1 a x b a linear equation = = How do we solve a set of them? 11 1 12 2 13 3 1n n 1 21 1 22 2 23 3 2n n 2 m1 1 m2 2 m3 3 mn n m a x a x a x ... a x b a x a x a x ... a x b ... . ........................................... = . .. a x a x a x ... a x b + + + + = + + + + = + + + + = This is an m n × system. If m n < , we will not have unique solutions. If m n , we may get into a situation inconsistency. If m n = , we may get unique set of solutions. Some observation: Suppose, the equation is 0 L : 2x 3y 5 - = Nothing is changed if we multiply or divide this equation by a constant.
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Therefore, 0 k L × can be used to replace 0 L . We denote this as 0 0 kL L Given two linear equations i L and j L , we can form any combination of these two to replace, say, j L . Thus, i j j kL L L + Interchanging the positions of two linear equations does not change anything. i j L L These operations that leave the system of equations invariant are called elementary operations. 2. Consider solution of linear equations. a. One unknown, one solution. If we have ax b, = we have a solution b x , if a 0 a = b. Two unknown, two solutions. Assume, 2x 3y 5 3x y 2 + = - = we have here x 1,y 1 = =
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If one equation is the multiple of another equation, we have infinite solutions. e.g. 2x 3y 5 4x 6 y 10 + = + = Geometrically, the two lines are parallel. The equations 11 1 12 2 1 21 1 22 2 2 a x a x b a x a x b + = + = have unique solutions if the determinant of this equation 11 12 11 22 12 21 21 22 a a | A| a a a a 0 a a = = - The determinant of a 3 3 × matrix could also be computed in a similar manner. For instance, 11 12 13 22 23 21 23 21 22 23 11 12 32 33 31 33 31 32 33 21 22 13 31 32 a a a a a a a | A| a a a a a a a a a a a a a a a a a = = × - × + ×
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The determinant | A| could be expressed as an expansion n n i j ij ij ij ij j 1 j 1 | A| a A ( 1) a M + = = = = - Where ij ij M minor A cofactor = = A minor ij M is obtained from the original matrix the resultant submatrix obtained by leaving i th row and j th column behind. 3. Systems in triangular and echelon form: An equation is in a triangular form if it can be expressed, via appropriate elementary operations, like below: 1 2 3 4 2 3 4 3 4 4 2x x 3x 2x 4 5x 6x x 0 2x 3x 1 2x 2 + - + = + + = - = = - This is easy to solve. A set of equations is in echelon form if (a) no equation is degenerate, (b) the leading term in an equation begins at the right of the leading term of the previous equation. The leading unknowns are called pivot element.
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Theorem. Suppose a set of m equations is in echelon form with n unknowns. If m n = , we have a unique set of solutions. If m n < , we can assign arbitrary values (parameters) for ( n m ) - variables, and solve the remaining m variables in terms of these parameters. See parametric form/free-variable form. Boils down
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topic3 - Systems of Linear Equations 1. Solve a x i =1 n ii...

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