Exam 8: The Geometry of Vector Spaces

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

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: $\left( 0,0 , \frac { 3 } { 8 } , \frac { 5 } { 8 } \right)$

Determine whether the point p is in the convex hull of S. - S= ,,, = 2 2 1 ,= 0 3 -1 ,= 1 8 -6 ,= -1 2 4 ,= 1 3 0

Provide an appropriate response. -Pick a set $S$ of four distinct points in $R ^ { 3 }$ such that aff $S$ is the plane $2 x _ { 1 } + x _ { 2 } - 3 x _ { 3 } = 14$ .

Provide an appropriate response -Let $\mathrm { p } _ { 0 } , \mathrm { p } _ { 1 }$ , and $\mathrm { p } _ { 2 }$ be points in $\mathfrak { R } ^ { \mathrm { n } }$ and define $\mathrm { f } _ { 0 } ( \mathrm { t } ) = ( 1 - \mathrm { t } ) \mathrm { p } _ { 0 } + \mathrm { tp } _ { 1 } , \mathrm { f } _ { 1 } ( \mathrm { t } ) = ( 1 - \mathrm { t } ) \mathrm { p } _ { 1 } + \mathrm { tp } _ { 2 }$ , and $g ( t ) = ( 1 - t ) f _ { 0 } ( t ) + t f _ { 1 } ( t )$ for $0 \leq t \leq 1$ . Find $\left. g \frac { 1 } { 2 } \right)$ in terms of the three points.

Provide an appropriate response -Which of the following statements are true? I: If $\mathrm { F } _ { 1 }$ and $\mathrm { F } _ { 2 }$ are 5 dimensional flats in $R ^ { 7 }$ , then the dimension of $\mathrm { F } _ { 1 } \cap \mathrm { F } _ { 2 }$ could have a dimension of $6 .$ II: If $F _ { 1 }$ and $F _ { 2 }$ are strictly separated, then $F _ { 1 } \cap F _ { 2 } = \varnothing$ .

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. -Which of the following statements are true for the set $S = \left\{ \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { \mathrm { k } } \right\}$ in $\mathfrak { R } \mathrm { n } ^ { \text {? } }$ I: If $S$ is affinely independent, then a point $p$ in $R ^ { n }$ cannot have any barycentric coordinates determined by $\mathrm { S }$ that are equal to 0 . II: If $S$ is affinely independent, then a point $p$ in aff $S$ has a unique representation as an affine combination of $\mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { \mathrm { k } }$ .

Provide an appropriate response. -If a Bézier curve is translated, $\mathbf { x } ( \mathrm { t } ) + \mathbf { b }$ , will the new curve always be a Bézier curve as well?

Provide an appropriate response -Let $\mathrm { H }$ be the hyperplane through the three points $\left[ \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right] , \left[ \begin{array} { l } 3 \\ 3 \\ 2 \end{array} \right] , \left[ \begin{array} { r } 2 \\ 5 \\ - 2 \end{array} \right]$ . Is the point $\mathbf { v } = \left[ \begin{array} { l } 4 \\ 2 \\ 1 \end{array} \right]$ on the same side of $\mathrm { H }$ as the origin? Justify your answer.

Let H be the hyperplane through the points. Find a linear functional f and a real number d such that H = [f : d]. - $\left[\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right],\left[\begin{array}{l}2 \\ 3 \\ 1 \\ 2\end{array}\right],\left[\begin{array}{r}-1 \\ 1 \\ 2 \\ 1\end{array}\right],\left[\begin{array}{r}3 \\ 2 \\ -1 \\ 1\end{array}\right]$

Provide an appropriate response. -A quadratic Bézier curve is determined by 3 control points $\mathbf { p } _ { 0 } , \mathbf { p } _ { 1 }$ , and $\mathbf { p } _ { 2 }$ . The equation is $\mathbf { x } ( \mathrm { t } ) =$ $( 1 - t ) ^ { 2 } \mathbf { p } _ { 0 } + 2 t ( 1 - t ) \mathbf { p } _ { 1 } + \mathrm { t } ^ { 2 } \mathbf { p } _ { 2 }$ . Construct the quadratic Bézier basis matrix $\mathrm { M } _ { \mathrm { B } }$ for $\mathbf { x } ( \mathrm { t } )$ .

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: $\left( \frac { 1 } { 8 } , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 } \right)$

Write y as an affine combination of the other points listed. - $\mathbf { v } _ { 1 } = \left[ \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } 0 \\ 4 \\ - 2 \end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 1 \\ - 5 \\ 1 \end{array} \right] , \mathbf { y } = \left[ \begin{array} { r } 5 \\ - 23 \\ 16 \end{array} \right]$

Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points. - $\left[ \begin{array} { l } 4 \\8\end{array} \right] , \left[ \begin{array} { l } 8 \\4\end{array} \right] , \left[ \begin{array} { l } 5 \\7\end{array} \right]$

Provide an appropriate response -Which of the following statements are true? I: If A and B are convex sets then A + B is convex. II: A four dimensional polytope always has the same number of vertices and edges.

Provide an appropriate response. -If 2 Bézier curves are joined at the point $\mathbf { p } _ { 3 }$ what is necessary for $\mathrm { C } ^ { 1 }$ parametric continuity?

Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points. - $\left[ \begin{array} { r } 4 \\ 8 \\ - 1 \end{array} \right] , \left[ \begin{array} { r } - 8 \\ - 16 \\ 4 \end{array} \right] , \left[ \begin{array} { r } 0 \\ 1 \\ 12 \end{array} \right] , \left[ \begin{array} { r } - 16 \\ - 31 \\ 22 \end{array} \right]$

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

Provide an appropriate response - $\mathbf { p } = \frac { 1 } { 2 } \mathbf { v } _ { 1 } + \frac { 1 } { 8 } \mathbf { v } _ { 2 } + \frac { 1 } { 8 } \mathbf { v } _ { 3 } + \frac { 1 } { 4 } \mathbf { v } _ { 4 }$ and $2 \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } - 2 \mathbf { v } _ { 3 } - \mathbf { v } _ { 4 } = \mathbf { 0 }$ Use the procedure in the proof of Caratheodory's Theorem to express $\mathbf { p }$ as a convex combination of three of the $\mathbf { v } _ { \mathbf { i } }$ 's.

Provide an appropriate response -Which of the following statements are true? I: If $S$ is a nonempty set, then conv $\mathrm { S } \subseteq \mathrm { S }$ . II: If $S$ and $\mathrm { T }$ are convex sets, then $\mathrm { S } \cap \mathrm { T }$ is also convex.