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A math quiz focused on linear independence of vectors and solving systems of linear equations using matrix form. It includes defining linear independence, finding the reduced row echelon form (rref) of a matrix, and using rref to determine if a system of linear equations has a solution and finding that solution.
Typology: Exercises
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Math 205B Quiz 03 page 1 10/02/2009 Name
and label its column vectors as c 1 , c 2 ,... , c 5. It’s a fact that the RREF of
is
2A. Let b =
b 1
b 2
b 3
b 4
. Use the above information to decide what conditions, if any, b 1 , b 2 , b 3 , and b 4 must satisfy in order
for Ax = b to have a solution.
2B. Without doing any work, find an easy solution of Ax = c 3.
2C. Since Ax = c 3 does have a solution, the entries of c 3 must satisfy the conditions in (2A). Show that this is so.
3A. Is the set S = {c 1 , c 2 ,... , c 5 } linearly independent? Explain.
3B. Show that it is possible to write c 3 as a linear combination of the other four vectors. (Give an explicit LC).
3C. Explain why it is impossible to write c 2 as a linear combination of the other four vectors.