a. Show that the set S is affinely independent.b. Find the

Chapter 8, Problem 16E

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QUESTION:

Let \(\mathbf{v}_{1}=\left[\begin{array}{l}0 \\ 1\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{l}1 \\ 5\end{array}\right], \quad \mathbf{v}_{3}=\left[\begin{array}{l}4 \\ 3\end{array}\right], \quad \mathbf{p}_{1}=\left[\begin{array}{l}3 \\ 5\end{array}\right]\), \(\mathbf{p}_{2}=\left[\begin{array}{l}5 \\ 1\end{array}\right], \quad \mathbf{p}_{3}=\left[\begin{array}{l}2 \\ 3\end{array}\right], \quad \mathbf{p}_{4}=\left[\begin{array}{r}-1 \\ 0\end{array}\right], \quad \mathbf{p}_{5}=\left[\begin{array}{l}0 \\ 4\end{array}\right]\), \(\mathbf{p}_{6}=\left[\begin{array}{l}1 \\ 2\end{array}\right], \mathbf{p}_{7}=\left[\begin{array}{l}6 \\ 4\end{array}\right] \text {, and } S=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}\).

a. Show that the set S is affinely independent.

b. Find the barycentric coordinates of \(\mathbf{p}_{1}, \mathbf{p}_{2} \text {, and } \mathbf{p}_{3}\) with respect to S.

c. On graph paper, sketch the triangle T with vertices \(v_1\), \(v_2\), and \(v_3\), extend the sides as in Figure 8, and plot the points \(\mathbf{p}_{4}, \mathbf{p}_{5}, \mathbf{p}_{6} \text {, and } \mathbf{p}_{7}\). Without calculating the actual values, determine the signs of the barycentric coordinates of points \(\mathbf{p}_{4}, \mathbf{p}_{5}, \mathbf{p}_{6} \text {, and } \mathbf{p}_{7}\).

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QUESTION:

Let \(\mathbf{v}_{1}=\left[\begin{array}{l}0 \\ 1\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{l}1 \\ 5\end{array}\right], \quad \mathbf{v}_{3}=\left[\begin{array}{l}4 \\ 3\end{array}\right], \quad \mathbf{p}_{1}=\left[\begin{array}{l}3 \\ 5\end{array}\right]\), \(\mathbf{p}_{2}=\left[\begin{array}{l}5 \\ 1\end{array}\right], \quad \mathbf{p}_{3}=\left[\begin{array}{l}2 \\ 3\end{array}\right], \quad \mathbf{p}_{4}=\left[\begin{array}{r}-1 \\ 0\end{array}\right], \quad \mathbf{p}_{5}=\left[\begin{array}{l}0 \\ 4\end{array}\right]\), \(\mathbf{p}_{6}=\left[\begin{array}{l}1 \\ 2\end{array}\right], \mathbf{p}_{7}=\left[\begin{array}{l}6 \\ 4\end{array}\right] \text {, and } S=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}\).

a. Show that the set S is affinely independent.

b. Find the barycentric coordinates of \(\mathbf{p}_{1}, \mathbf{p}_{2} \text {, and } \mathbf{p}_{3}\) with respect to S.

c. On graph paper, sketch the triangle T with vertices \(v_1\), \(v_2\), and \(v_3\), extend the sides as in Figure 8, and plot the points \(\mathbf{p}_{4}, \mathbf{p}_{5}, \mathbf{p}_{6} \text {, and } \mathbf{p}_{7}\). Without calculating the actual values, determine the signs of the barycentric coordinates of points \(\mathbf{p}_{4}, \mathbf{p}_{5}, \mathbf{p}_{6} \text {, and } \mathbf{p}_{7}\).

ANSWER:

Solution 16Ea)Suppose the set, , where The objective is to show that is affinely independent.Compute the following points,Suppose there exist a scalar such that That is Equati

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