# Worksheet 6-3: Basic Feasible Solutions

Download: [CMSE382-WS6_3.pdf](CMSE382-WS6_3.pdf)

```{warning}
This is an AI-generated transcript of the worksheet and may contain errors or inaccuracies. Please refer to the original course materials for authoritative content.
```

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## Worksheet 6-3: Q1

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

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

(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

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

(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.

```{image} ../../../figures/WS_convex_hull_problem.png
:width: 500px
:align: center
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(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.