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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 rowequivalent 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 + 1  2 1 + 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 = 1  2 1...
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 Winter '08
 chuchel
 Linear Algebra, Algebra, Equations

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