Exam 8: The Geometry of Vector Spaces

Provide an appropriate response -Which of the following statements are true? I: A polytope is the affine hull of a finite set of points. II: An extreme point of a polytope P is any point in the convex hull of 2 vertices.

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 if $\mathbf { p }$ is a point on the line through $\mathbf { a }$ and $\mathbf { b }$ ? I: $\mathbf { p }$ is an affine combination $\mathbf { a }$ and $\mathbf { b }$ . II: $\widetilde { \mathbf { a } } , \widetilde { \mathbf { b } }$ , and $\tilde { \mathbf { p } }$ are linearly independent. of $\text { [a b p] is } 0 \text {. }$ III: The determinant of is 0 .

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

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. -Consider the affinely independent set $S = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ in $R ^ { 2 }$ . These points form a triangular region. If the barycentric coordinates of a point $\mathbf { p }$ are all positive $( + , + , + )$ , then where is $\mathbf { p }$ with respect to the triangle?

Provide an appropriate response. -Suppose $\left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ is a basis for $R ^ { 3 }$ . Which of the following statements are true? I: Span $\left\{ \mathbf { v } _ { 2 } - \mathbf { v } _ { 1 } , \mathbf { v } _ { 3 } - \mathbf { v } _ { 1 } \right\}$ is a plane in $\mathscr { R } ^ { 3 }$ . II: Aff $\left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ is the plane through $\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }$ , and $\mathbf { v } _ { 3 }$ .

Determine whether the point p is in the convex hull of S. - = ,,, = 2 0 5 1 ,= 1 1 -1 -3 ,= 3 2 0 -4 ,= 0 -1 4 2

Determine if the vector p is in Span S or aff S. -Let $\mathbf { v } _ { 1 } = \left[ \begin{array} { r } 2 \\ 1 \\ 3 \\ - 2 \end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } 3 \\ - 1 \\ 0 \\ 4 \end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 5 \\ 4 \\ - 1 \\ - 2 \end{array} \right] , \mathbf { p } = \left[ \begin{array} { r } 2 \\ - 3 \\ 7 \\ 2 \end{array} \right]$ , and $\mathrm { S } = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ . It can be shown that $S$ is linearly independent.

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. - $S = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } , \mathbf { v } _ { 4 } \right\}$ Barycentric coordinates: $( 4 , - 2,0 , - 1 )$

Provide an appropriate response -Which of the following statements are true? I: The open ball $\mathrm { B } ( \mathbf { p } , \delta )$ is a convex set. II: The convex hull of a compact set is compact.

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 $\mathfrak { R } ^ { n }$ ? I: $\left\{ \mathbf { v } _ { 2 } - \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { \mathrm { k } } - \mathbf { v } _ { 1 } \right\}$ is linearly dependent if and only if $S$ is affinely dependent. II: If $S$ is affinely independent and a point $p$ in $R ^ { n }$ has all positive barycentric coordinates determined by $S$ , then $p$ is not in aff $S$ .

Provide an appropriate response. -Which of the following statements are true? I: $2 \mathbf { v } _ { 1 } + 2 \mathbf { v } _ { 2 } - 3 \mathbf { v } _ { 3 }$ is an affine combination of the 3 vectors. II: The affine hull of two distinct points is a plane. III: If $S = \{ x \}$ , then aff $S = \{ x \}$ . IV: If a set of vectors in $\mathfrak { R } ^ { \mathrm { n } }$ is linearly independent, then every vector in $\mathfrak { R } ^ { \mathrm { n } }$ can be written as an affine combination of these vectors.

Provide an appropriate response -Consider the set $S$ of points $\left[ \begin{array} { l } x \\ y \end{array} \right]$ in $R ^ { 2 }$ such that $y = \frac { 1 } { x }$ and $x \geq \frac { 1 } { 2 }$ . Are the sets $S$ and conv $S$ both closed?

Provide an appropriate response -A five dimensional hypercube $C ^ { 5 }$ has how many 2 -faces ?

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it. -Consider the affinely independent set $\mathrm { S } = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ in $\mathfrak { R } ^ { 2 }$ . These points form a triangular region. If the barycentric coordinates of a point $\mathbf { p }$ are $( + , 0 , + )$ , then where is $p$ with respect to the triangle?

Provide an appropriate response -A four dimensional simplex $S ^ { 4 }$ has how many 2 -faces?

Provide an appropriate response. - $\mathbf { v } _ { 1 } = \left[ \begin{array} { l } 2 \\ 2 \\ 5 \end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { l } 2 \\ 4 \\ 3 \end{array} \right]$ and $S = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ . Aff $S$ is a plane in $R ^ { 3 }$ . Give its equation.

Provide an appropriate response. - $[ ( \operatorname { aff } A ) \cup ( \operatorname { aff } B ) ] \subset ( A \cup B )$ What property must the set $( A \cup B )$ have if the above statement is true?