Exam 8: The Geometry of Vector Spaces

Sketch the graph of the convex hull of S. -In $R ^ { 2 } , S$ is the set of points $\left[ \begin{array} { l } x \\ y \end{array} \right]$ where $y = x ^ { 2 }$ and $x \geq 0$

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. -Let $\mathbf { p }$ be a point in the interior of $\triangle \mathrm { abc }$ with barycentric coordinates $\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } , \frac { 1 } { 8 } \right)$ . What is the area of $\Delta$ pbc with respect to the area of $\Delta \mathbf { a b c }$ ?

Provide an appropriate response. -Which of the following statements are true? I: Suppose $\mathrm { f } : \mathfrak { R } ^ { \mathrm { n } } \rightarrow R ^ { \mathrm { m } }$ is a linear transformation and $S$ is an affine subset of $R ^ { \mathrm { n } }$ . It follows that the set of images $f ( S )$ is an affine subset of $R ^ { m }$ . II: If $A \subset B$ , then aff $A \subset B$ .

Write y as an affine combination of the other points listed. - $\mathbf { v } _ { 1 } = \left[ \begin{array} { r } 3 \\- 2\end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } 1 \\- 1\end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 9 \\- 1\end{array} \right] , \mathbf { y } = \left[ \begin{array} { l } 7 \\4\end{array} \right]$

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. - $\left[ \begin{array} { l } 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 5 \end{array} \right] , \left[ \begin{array} { l } 0 \\ 6 \end{array} \right] , \mathrm { p } = \left[ \begin{array} { l } 2 \\ 3 \end{array} \right]$

Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge of conv S which is a tetrahedron. - $\mathrm { S } = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } , \mathbf { v } _ { 4 } , \right\}$ Barycentric coordinates: $\left( \frac { 1 } { 2 } , 0 , \frac { 1 } { 8 } , \frac { 3 } { 8 } \right)$

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. - $\left[ \begin{array} { r } 1 \\ - 1 \\ 2 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 2 \\ 1 \\ 0 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 1 \\ 2 \\ - 2 \\ 0 \end{array} \right] , \mathbf { p } = \left[ \begin{array} { r } - 1 \\ - 11 \\ 14 \\ 3 \end{array} \right]$

Provide an appropriate response -Let (pos S)be the set of all positive combinations of the points of S. Which of the following statements are true? I: $( \operatorname { pos } S ) \cap ($ aff $S ) =$ conv $S$ II: If a unique linear combination of points is both positive and affine, then it must be convex.

Sketch the graph of the convex hull of S. - $\operatorname{In} \mathfrak{R}^{2}, \text { let } \mathrm{S}=\left\{\left[\begin{array}{l}0 \\3\end{array}\right]\right\} \cup\left\{\left[\begin{array}{l}\mathrm{x} \\0\end{array}\right]: 0 \leq \mathrm{x} \leq 4\right\}$

Provide an appropriate response -Let int S be the set of all interior points of S, and let cl S be the closure of S (S ∪ the set of all boundary points of S). Which of the following statements are true? I: If S is convex, then int S is convex. II: If S is convex, then cl S is convex.

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. -If $\left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } \right\}$ in $R ^ { \mathbf { n } }$ is affinely dependent, what is known about the 2 points?

Provide an appropriate response -Which of the following statements are true? I: A linear transformation from $\mathscr { R }$ to $R ^ { \mathrm { n } }$ is called a linear functional. II: The convex hull of a closed set is closed.

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. -How many points need to be in a set in $R ^ { 6 } 6$ to guarantee that the set is affinely dependent?

Provide an appropriate response. -How many control points are needed for a cubic Bézier curve?

Provide an appropriate response -Which of the following statements are true? I: If $S = \{ ( x , y ) : x - y = 0$ and $x \geq 0 \}$ and if $P$ is its profile, then conv $P = S$ . II: If $S = \{ ( x , y ) : x - y = 0$ and $0 \leq x \leq 5 \}$ and if $P$ is its profile, then conv $P = S$ .

Provide an appropriate response -Which of the following statements are true? I: If $A \subset B$ , then $A \subset \operatorname { conv } B$ . II: If $A \subset B$ , then conv $A \subset c o n v B$ .

Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d]. - $\left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] , \left[ \begin{array} { r } 3 \\ - 1 \\ 5 \end{array} \right] , \left[ \begin{array} { r } 4 \\ 2 \\ - 2 \end{array} \right]$

Provide an appropriate response. -Let $x ( t )$ be a Bézier curve and the tangent vector $x ^ { \prime } ( t )$ is computed. What does knowing that $\mathbf { x } ^ { \prime } ( 0 ) =$ $3 \left( \mathbf { p } _ { 1 } - \mathrm { p } _ { 0 } \right)$ tell you?

Determine if the vector p is in Span S or aff S. - $\text { Let } \mathbf{v}_{1}=\left[\begin{array}{r}2 \\1 \\3 \\-2\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}3 \\-1 \\0 \\4\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}5 \\4 \\-1 \\-2\end{array}\right], p=\left[\begin{array}{r}5 \\-3 \\5 \\3\end{array}\right] \text {, and } \mathrm{S}=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\} \text {. It can be shown that } \mathrm{S} \text { is linearly }$ independent.

Determine if the vector p is in Span S or aff S. -Let $\mathbf { v } _ { 1 } = \left[ \begin{array} { r } 2 \\ 1 \\ 3 \\ - 2 \end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } 3 \\ - 1 \\ 0 \\ 4 \end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 5 \\ 4 \\ - 1 \\ - 2 \end{array} \right] , \mathbf { p } = \left[ \begin{array} { r } 10 \\ 12 \\ - 6 \\ - 8 \end{array} \right]$ , and $\mathrm { S } = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ . It can be shown that $\mathrm { S }$ is linearly independent.