mws_gen_sle_bck_system - Chapter 04.05 System of Equations...

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04.05.1 Chapter 04.05 System of Equations After reading this chapter, you should be able to: 1. setup simultaneous linear equations in matrix form and vice-versa, 2. understand the concept of the inverse of a matrix, 3. know the difference between a consistent and inconsistent system of linear equations, and 4. learn that a system of linear equations can have a unique solution, no solution or infinite solutions. Matrix algebra is used for solving systems of equations. Can you illustrate this concept? Matrix algebra is used to solve a system of simultaneous linear equations. In fact, for many mathematical procedures such as the solution to a set of nonlinear equations, interpolation, integration, and differential equations, the solutions reduce to a set of simultaneous linear equations. Let us illustrate with an example for interpolation. Example 1 The upward velocity of a rocket is given at three different times on the following table. Table 5.1. Velocity vs. time data for a rocket Time, t Velocity, v (s) (m/s) 5 106.8 8 177.2 12 279.2 The velocity data is approximated by a polynomial as  12. t 5 , 2 c bt at t v Set up the equations in matrix form to find the coefficients c b a , , of the velocity profile. Solution The polynomial is going through three data points     3 3 2 2 1 1 , t and , , , , v v t v t where from table 5.1. 8 . 106 , 5 1 1 v t 2 . 177 , 8 2 2 v t
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04.05.2 Chapter 04.05 2 . 279 , 12 3 3 v t Requiring that  c bt at t v 2 passes through the three data points gives  c bt at v t v 1 2 1 1 1  c bt at v t v 2 2 2 2 2  c bt at v t v 3 2 3 3 3 Substituting the data    3 3 2 2 1 1 , and , , , , v t v t v t gives    8 . 106 5 5 2 c b a    2 . 177 8 8 2 c b a    2 . 279 12 12 2 c b a or 8 . 106 5 25 c b a 2 . 177 8 64 c b a 2 . 279 12 144 c b a This set of equations can be rewritten in the matrix form as 2 . 279 2 . 177 8 . 106 12 144 8 64 5 25 c b a c b a c b a The above equation can be written as a linear combination as follows 2 . 279 2 . 177 8 . 106 1 1 1 12 8 5 144 64 25 c b a and further using matrix multiplication gives 2 . 279 2 . 177 8 . 106 1 12 144 1 8 64 1 5 25 c b a The above is an illustration of why matrix algebra is needed. The complete solution to the set of equations is given later in this chapter. A general set of
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.

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mws_gen_sle_bck_system - Chapter 04.05 System of Equations...

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