mat22a-midterm1-sol

# mat22a-midterm1-sol - Math 22A UC Davis, Winter 2011 Prof....

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Math 22A UC Davis, Winter 2011 Prof. Dan Romik Midterm Exam 1 Solutions 1. Find the general form of the solution of the linear system 2 x - 2 y - 4 z - 6 w = - 2 - x + y + 2 z + 3 w = 1 5 x - 4 y + 4 z + 4 w = 1 4 x - 3 y + 6 z + 7 w = 2 Solution. We encode the system as the augmented matrix 2 - 2 - 4 - 6 - 2 - 1 1 2 3 1 5 - 4 4 4 1 4 - 3 6 7 2 Using the Gaussian elimination algorithm, we can perform a sequence of ele- mentary row operations to bring the matrix to its reduced row echelon form. This results in the equivalent augmented matrix 1 0 12 16 5 0 1 14 19 6 0 0 0 0 0 0 0 0 0 0 Here z and w are the independent (non-pivot) variables, so we denote z = s and w = t where s,t are parameters taking arbitrary real values. Therefore the general solution to the system is of the form x = 5 - 12 s - 16 t, y = 6 - 14 s - 19 t, z = s, w = t, or in vector notation x y z w = 5 6 0 0 + s

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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-midterm1-sol - Math 22A UC Davis, Winter 2011 Prof....

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