AE202 Lecture 4 Linear Systems of Equations

AE202 Lecture 4 Linear Systems of Equations - AE202...

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AE202 Aerospace Flight Mechanics Lecture 4: Linear Systems and Matrix Operations
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Linear systems Linear systems of equations have many applications in engineering – Structural analysis • Finite Element analysis – Electrical networks – Control engineering
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Solving Systems of Linear Equations Linear system of equations To solve in MATLAB consider the matrix form – where • A : m × n matrix • b : n × 1 vector • x : m × 1 vector of unknowns • m : number of equations • n: number of variables 3 6 2 2 4 2 3 36 x y z x y z xy      Ax b
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Solving Systems of Linear Equations etc. >> A=[-3 6 -1;2 -4 -2; 1 1 3] >> b=[-2;3;6] >> x=A\b Answer: >> x = 5.5417 2.3333 -0.6250 Does answer make sense ?
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Problems with Linear equations For there to exist a unique solution to the linear system – All the equations must be independent • none of the equations can be derived from the others • e.g. and are not independent – All the equations must be consistent • The equations must have a common solution • e.g. and are inconsistent – All the rows of matrix A must be linearly independent. • Matrix A must be full rank ie. det(A)!=0 Ax b 2 3 4 6 3 9 12 5 x y z x y z x y z        5 2 3 4 4 2 9 x y z x y z x y z       2 4 4 2 4 7 5 x y z x y z x y z       5 2 3 4 426 x y z x y z x y z      
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Computational Considerations Does the equation have a unique solution? – If so, how would you compute the solution? or – More work to compute the inverse first and multiply and can be less accurate if inverse is stored using a finite number of digits Same is true for matrices. To Solve – Could solve – Matrix inverses are generally computationally expensive to compute 13 39 x 39/13 x   13 *39 x inv 1 x A b Ax b
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Computational Considerations cont.
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AE202 Lecture 4 Linear Systems of Equations - AE202...

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