Math 171A Homework Assignment # 2
Instructor: Jiawang Nie
Assigned Date: January 13, 2010 Due Date: January 20, 2010
1. (8 points) Consider the feasible set Fdefined by the following constraints
x1+x2≥4, x1+ 3x2≥6,6x1−x2≤18,3≤x2≤6, x1≥ −1.
(a) Express Fin the standard form Ax ≥b. Write down Aand bexplicitly.
(b) Draw the set Fgraphically in the plane, and find all the corner points of F.
(c) Solve the following LP
minimize 2x1−3x2
subject to Ax ≥b
where Aand bare from part (a).
(d) Compute the residual vector r(x) for all the constraints at the point ¯x= (2,4), and find
the constraints whose residuals would decrease after a positive step αalong the direction
p= (1,−2) emanating from the point ¯x.
2. (8 points) Consider the linear system Ax =bwhere
A=
−3−2−1−2−6
−3 3 −6 0 −3
−6−6 0 −1 0
, b =
−6
0
−5
.
(a) Determine the rank of A. Is the system Ax =bcompatible?
(b) List all the submatrices of Athat have the same rank as A.
(c) Find two distinct basic solutions (along with the corresponding basic sets), and verify
that Ax is equal to b in each case.
(d) How many basic solutions does this linear system have?
3. (4 points) Consider the linear system Ax =bwhere
A=
3 3 6
2−3 6
1 6 0
2 0 1
6 3 0
, b =
4
3
1
−1
−5
.
(a) Determine the rank of A and check if Ax =bis compatible.
(b) Find two basic sets (each specifying a linearly independent subset of the rows of A), and
use them to solve Ax =b. Comment on your results.
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