mat22a-final-practice-sol

mat22a-final-practice-sol - Math 22A UC Davis Winter 2011...

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Unformatted text preview: Math 22A UC Davis, Winter 2011 Prof. Dan Romik Solutions to practice questions for the final 1. You are given the linear system of equations 2 x 1 + 4 x 2 + x 3 + x 4 = 8 x 1 + 2 x 2 + x 3 = 5- x 1- 2 x 2 + x 3- 2 x 4 =- 1 x 1 + 2 x 2 + x 4 = 3 (a) Write an augmented matrix representing the system. Solution. 2 4 1 1 8 1 2 1 5- 1- 2 1- 2- 1 1 2 0 1 3 (b) Find a reduced row echelon form (RREF) matrix that is row-equivalent to the augmented matrix. Solution. 1 2 0 1 3 0 0 1- 1 2 0 0 0 0 0 0 (c) Find the general solution of the system. Solution. x 1 x 2 x 3 x 4 = 3 2 + λ 1 - 2 1 + λ 2 - 1 1 1 (d) Write the homogeneous system of equations associated with the above (nonhomogeneous) system and find its general solution. 1 Solution. The homogeneous system is represented by the augmented matrix 2 4 1 1 1 2 1- 1- 2 1- 2 1 2 0 1 This is the same as the original system except that the rightmost column is the zero vector. The equivalent RREF is therefore 1 2 0 1 0 0 1- 1 0 0 0 0 0 0 and the general solution is x 1 x 2 x 3 x 4 = λ 1 - 2 1...
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This note was uploaded on 11/13/2011 for the course MATH 22a taught by Professor Chuchel during the Winter '08 term at UC Davis.

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mat22a-final-practice-sol - Math 22A UC Davis Winter 2011...

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