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 equationsx1+ 3x2− 5x3= 4x1+ 4x2− 8x3= 7−3x1− 7x2+ 9x3= −6Write your answer in parametric vector form.We create the augmented matrix1 3 −5 41 4 −8 7−3 −7 9 −6Row operations will eventually give1 0 4 −50 1 −3 30 0 0 0So the solution is x1= −5 − 4x3, x2= 3 + 3x3, x3is free. Written in vector form, thisisx1x2x3=−530+ t−4312. Let u =41−4and 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 2So the system is consistent and thus u is in the span of the columns of A.(b) (4 pts) For which vectors b =b1b2b3is 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− 2b2Since 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
View Full Document