Worksheet 6-3: Basic Feasible Solutions

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Worksheet 6-3: Q1

Consider the linear system \(A\mathbf{x} = \mathbf{b}\) given by

\[\begin{split}\begin{bmatrix} 1 & 5 & 3 & 4 & 6 \\ 0 & 1 & 3 & 5 & 6 \end{bmatrix} \mathbf{x} = \begin{bmatrix} 14 \\ 7 \end{bmatrix}\end{split}\]

(a) Which subsets of 2 columns of \(A\) are linearly independent? (Hint: there are 10 possible pairs; identify the one that is not linearly independent.)

(b) Consider the point \(\mathbf{x} = (0, 2, 0, 1, 0)^\top\). Verify that this is a solution to \(A\mathbf{x} = \mathbf{b}\).

(c) Is \(\mathbf{x} = (0, 2, 0, 1, 0)^\top\) a basic feasible solution? Why or why not?

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Worksheet 6-3: Q2

Consider the linear system \(A\mathbf{x} = \mathbf{b}\) given by

\[\begin{split}\begin{bmatrix} 1 & 7 & 22 & 0 & 5 & 0 & 1 \\ 0 & 3 & -5 & 0 & 2 & 1 & 2 \\ 0 & 1 & 0 & 1 & -4 & 0 & 3 \end{bmatrix} \mathbf{x} = \begin{bmatrix} 7 \\ 1 \\ 2 \end{bmatrix}\end{split}\]

(a) Check that the 1st, 4th, and 6th columns of \(A\) are linearly independent.

(b) Find the solution to \(A\mathbf{x} = \mathbf{b}\) with nonzero entries only in positions 1, 4, and 6. Is it a basic feasible solution?

(c) Check that the 1st, 3rd, and 4th columns of \(A\) are linearly independent.

(d) Find the solution to \(A\mathbf{x} = \mathbf{b}\) with nonzero entries only in positions 1, 3, and 4. Is it a basic feasible solution? Why or why not?

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Worksheet 6-3: Q3

Let \(S\) be a closed, bounded, and convex set. The figure below (Fig. a) shows a sampling of points from \(S\) that includes all the extreme points and some interior points.

(a) Mark the extreme points \(\text{ext}(S)\) on Fig. b.

(b) Highlight the region \(\text{conv}(\text{ext}(S))\) on Fig. c.

(c) What is your best guess for the full set \(S\)? Explain your answer.