Question 1

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.

Accepted Answer

A) Neither I or II 
B) II 
C) I 
D) Both I and II

Question 2

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 .

Accepted Answer

A)  I and II 
B)  I and III 
C)  II 
D)  None are true.

Question 3

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?

Accepted Answer

A)  Both tangent vectors at $p _ { 3 }$ need to be equal. 
B)  Both tangent vectors at $\mathbf { p } _ { 3 }$ need to point in the same direction. 
C)  The second derivative at $\mathbf { p } _ { 3 }$ needs to equal 0 . 
D)  $\mathbf { p } _ { 3 }$ just needs to be a control point for both curves.

Question 4

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?

Accepted Answer

A)  Outside 
B)  Vertex 
C)  Edge 
D)  Inside

Question 5

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 $\mathbf { v } _ { 3 }$.

Accepted Answer

A)  II 
B)  I 
C)  Both I and II 
D)  Neither I or II

Question 6

Determine whether the point p is in the convex hull of S.
-$$\begin{array} { l } 
\mathrm { S } = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } , \mathbf { v } _ { 4 } \right\} \
\mathbf { v } _ { 1 } = \left[ \begin{array} { l } 
2 \
0 \
5 \
1
\end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } 
1 \
1 \
- 1 \
- 3
\end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 
3 \
2 \
0 \
- 4
\end{array} \right] , \mathbf { p } = \left[ \begin{array} { r } 
0 \
- 1 \
4 \
2
\end{array} \right]
\end{array}$$

Accepted Answer

A)  $\mathrm { p } \in \mathrm { conv } \mathrm { S } . \mathrm { p } = \frac { 3 } { 8 } \mathbf { v } _ { 1 } + \frac { 1 } { 4 } \mathbf { v } _ { 2 } + \frac { 3 } { 8 } \mathbf { v } _ { 3 }$ 
B)  $\mathbf { p } \in$ conv S. $\mathbf { p } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } - \mathbf { v } _ { 3 }$ 
C)  $\mathbf { p } \notin \operatorname { conv } \mathrm { S }$. 
D)  $\mathbf { p } \in$ conv S. $\mathbf { p } = 2 \mathbf { v } _ { 1 } + 3 \mathbf { v } _ { 2 } - 4 \mathbf { v } _ { 3 }$

Question 7

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.

Accepted Answer

A)  $\mathbf { p } \in$ span $S$ and $\mathbf { p } \in$ aff $S$ 
B)  $\mathbf { p } \notin \operatorname { span } S$ and $\mathbf { p } \in$ aff $S$ 
C)  $\mathbf { p } \notin$ span $S$ and $\mathbf { p } \notin$ aff $S$ 
D)  $\mathbf { p } \in$ span $S$ and $\mathbf { p } \notin$ aff $S$

Question 8

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: $( 4 , - 2,0 , - 1 )$

Accepted Answer

A)  Face 
B)  Outside 
C)  Inside 
D)  Edge

Question 9

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.

Accepted Answer

A)  Neither I or II 
B)  II 
C)  I 
D)  Both I and II

Question 10

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 $S$.

Accepted Answer

A)  I 
B)  II 
C)  Neither I or II 
D)  Both I and II

Question 11

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.

Accepted Answer

A)  II and IV 
B)  I and II 
C)  I and III 
D)  I, III, and IV

Question 12

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?

Accepted Answer

A)  No. Neither set is closed. 
B)  No. S is closed but conv $S$ is not. 
C)  Yes. They are both closed sets. 
D)  No. Conv S is closed but $\mathrm { S }$ is not.

Question 13

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

Accepted Answer

A)  24 
B)  32 
C)  80 
D)  40

Question 14

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?

Accepted Answer

A)  Vertex 
B)  Edge 
C)  Outside 
D)  Inside

Question 15

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

Accepted Answer

A)  15 
B)  10 
C)  5 
D)  4

Question 16

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.

Accepted Answer

A)  $13 x + 14 y = 17$ 
B)  $x = 2$ 
C)  $x + y + z = 0$ 
D)  $6 x + 9 y + 7 z = 0$

Question 17

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?

Accepted Answer

A)  There is nothing special about $( \mathrm { A } \cup \mathrm { B } )$. The statement is always true. 
B)  $( A \cup B )$ must contain collinear points. 
C)  $\mathrm { A } \subset \mathrm { B }$ so that $( \mathrm { A } \cup \mathrm { B } ) = \mathrm { A }$. 
D)  $( A \cup B )$ has to be affine.

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.

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?

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.

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 ?

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

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?

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.

Provide an appropriate response. -If 2 Bézier curves are joined at the point p3\mathbf { p } _ { 3 }p3​ what is necessary for G1\mathrm { G } ^ { 1 }G1 geometric continuity?

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

Provide an appropriate response -Consider the set SSS of points [xy]\left[ \begin{array} { l } x \\ y \end{array} \right][xy​] in R2R ^ { 2 }R2 such that y=1xy = \frac { 1 } { x }y=x1​ and x≥12x \geq \frac { 1 } { 2 }x≥21​ . Are the sets SSS and conv SSS both closed?

Provide an appropriate response -A five dimensional hypercube C5C ^ { 5 }C5 has how many 2 -faces ?

Provide an appropriate response -A four dimensional simplex S4S ^ { 4 }S4 has how many 2 -faces?

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