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Berkeley MATH 54 - Quiz 2

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Name:Section: 9-10 11-12 2-3Math 54 Quiz 2February 4, 2008GSI: Rob BayerYou have twenty minutes to complete this quiz. You must show your work.1. (4 pts) Find the general solution to the system of equationsx1+ 3x2− 5x3= 4x1+ 4x2− 8x3= 7−3x1− 7x2+ 9x3= −6Write your answer in parametric vector form.We create the augmented matrix1 3 −5 41 4 −8 7−3 −7 9 −6Row operations will eventually give1 0 4 −50 1 −3 30 0 0 0So the solution is x1= −5 − 4x3, x2= 3 + 3x3, x3is free. Written in vector form, thisisx1x2x3=−530+ t−4312. Let u =41−4and A =1 0 −40 3 −2−2 6 3(a) (2 pts) Is u in the span of the columns of A?Asking if u is in the span of the columns of A is the same as asking if Ax = uhas a solution, so we create an augmented matrix and check for consistency:1 0 −4 40 3 −2 1−2 6 3 −4→1 0 −4 40 3 −2 10 6 −5 4→1 0 −4 40 3 −2 10 0 −1 2So the system is consistent and thus u is in the span of the columns of A.(b) (4 pts) For which vectors b =b1b2b3is b in the span of the columns of A?This is the same as asking when Ax = b has a solution. We do the same steps asabove:1 0 −4 b10 3 −2 b2−2 6 3 b3→1 0 −4 b10 3 −2 b20 6 −5 b3+ 2b1→1 0 −4 b20 3 −2 b20 0 −1 b3+ 2b1− 2b2Since there is a pivot in every row, this system will always be consistent, nomatter what we choose for b1, b2, b3. Thus, all vectors are in the span of thecolumns of


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Berkeley MATH 54 - Quiz 2

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