Question 1

Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points.&#10;

Accepted Answer

The set is affinely dependent because the linear combination $3 \left[ \begin{array} { l } 4 \ 8 \end{array} \right] + \left[ \begin{array} { l } 8 \ 4 \end{array} \right] - 4 \left[ \begin{array} { l } 5 \ 7 \end{array} \right] = \left[ \begin{array} { l } 0 \ 0 \end{array} \right]$ holds, indicating a dependence relation.

Question 2

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

B

Question 3

Provide an appropriate response.&#10;

Accepted Answer

Since $\left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \right\}$ is a basis for $R ^ { 3 }$, it spans $R ^ { 3 }$. Therefore, any two vectors in $R ^ { 3 }$ will span a plane. Thus, statement I is true. Since $\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }$, and $\mathbf { v } _ { 3 }$ are not collinear, they do not lie in a plane. Thus, statement II is false.

Question 4

Provide an appropriate response.&#10;Which of the following statements are true?

Accepted Answer

I: This statement is false because an affine combination of vectors requires that the coefficients sum to 1, which is not guaranteed in the given expression.III: This statement is true because the affine hull of a single point is the point itself, as there are no other points to form a line or plane with.

Question 5

Provide an appropriate response.&#10;

Accepted Answer

The affine hull of a set, denoted as $\operatorname{aff}(A)$, is the smallest affine space that contains the set. The statement $[ ( \operatorname { aff } A ) \cup ( \operatorname { aff } B ) ] \subset ( A \cup B )$ would only be true if $A \cup B$ itself forms an affine space, meaning it must be closed under affine combinations. This is what being affine implies.

Question 6

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

I: The statement is true because affine dependence of a set $S$ means that there exists a non-trivial affine combination of the points in $S$ that equals zero. This directly translates to the linear dependence of the set $\left\{ \mathbf { v } _ { 2 } - \mathbf { v } _ { 1 } , \ldots , \mathbf { v } _ { \mathrm { k } } - \mathbf { v } _ { 1 } \right\}$, as the subtraction of one point from the others essentially checks for linear combinations that sum to the zero vector, indicating linear dependence.II: This statement is false because if $p$ has all positive barycentric coordinates with respect to $S$, it actually implies that $p$ lies inside the affine hull of $S$ (aff $S$). Barycentric coordinates are used to express a point as a combination of points that define a simplex, and having all positive coordinates means the point is strictly within the simplex formed by $S$, not outside of it.

Question 7

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

The answer of Find the barycentric coordinates of p with...

Question 8

Determine if the vector p is in Span S or aff S.&#10;

Accepted Answer

Since S is linearly independent, it forms a basis for Span S. Since p is a linear combination of v1, v2, and v3, p is in Span S. Since p is not a linear combination of v1, v2, and v3, p is not in aff S.

Question 9

Provide an appropriate response.&#10;

Accepted Answer

The vectors $\mathbf{v}_1$, $\mathbf{v}_2$, and $\mathbf{v}_3$ all have the same $x$-component, which is $2$. This means that all points in the set $S$ share the same $x$-coordinate, defining a plane parallel to the $yz$-plane at $x = 2$.

Question 10

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

Barycentric coordinates represent a point's position in relation to the vertices of a triangle. The given coordinates $(+, 0, +)$ indicate that $p$ lies on the line segment between two vertices (since one coordinate is zero), but the positive signs imply it is actually inside the triangle, influenced by the vertices associated with the positive values.

Question 11

Write y as an affine combination of the other points listed.&#10;

Accepted Answer

The answer of Write y as an affine combination of...

Question 12

Write y as an affine combination of the other points listed.&#10;

Accepted Answer

The answer of Write y as an affine combination of...

Question 13

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

The answer of Find the barycentric coordinates of p with...

Question 14

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

The answer of Find the barycentric coordinates of p with...

Question 15

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

The answer of Find the barycentric coordinates of p with...

Question 16

Determine if the vector p is in Span S or aff S.&#10;

Accepted Answer

The answer of Determine if the vector p is in...

Question 17

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 18

Determine if the set of points is affinely dependent. If so, construct an affine dependence relation for the points.&#10;

Accepted Answer

The answer of Determine if the set of points is...

Question 19

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 20

Determine if the vector p is in Span S or aff S.&#10;

Accepted Answer

The answer of Determine if the vector p is in...

Question 21

Provide an appropriate response&#10;Let (pos S)be the set of all positive combinations of the points of S. Which of the following statements are true?

Accepted Answer

The answer of Provide an appropriate response&#10;Let (pos S)be the...

Question 22

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 23

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 24

Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge&#10;of conv S which is a tetrahedron.&#10;

Accepted Answer

The answer of Use the barycentric coordinates with respect to...

Question 25

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

The answer of Find the barycentric coordinates of p with...

Question 26

Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge&#10;of conv S which is a tetrahedron.&#10;

Accepted Answer

The answer of Use the barycentric coordinates with respect to...

Question 27

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

The answer of Find the barycentric coordinates of p with...

Question 28

Determine whether the point p is in the convex hull of S.&#10;

Accepted Answer

The answer of Determine whether the point p is in...

Question 29

Provide an appropriate response&#10;Which of the following statements are true?

Accepted Answer

The answer of Provide an appropriate response&#10;Which of the following...

Question 30

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 31

Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge&#10;of conv S which is a tetrahedron.&#10;

Accepted Answer

The answer of Use the barycentric coordinates with respect to...

Question 32

Use the barycentric coordinates with respect to S to determine if the point p is inside, outside, on a face, or on the edge&#10;of conv S which is a tetrahedron.&#10;

Accepted Answer

The answer of Use the barycentric coordinates with respect to...

Question 33

Determine whether the point p is in the convex hull of S.&#10;

Accepted Answer

The answer of Determine whether the point p is in...

Question 34

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 35

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 36

Sketch the graph of the convex hull of S.&#10;

Accepted Answer

The answer of Sketch the graph of the convex hull...

Question 37

Sketch the graph of the convex hull of S.&#10;

Accepted Answer

The answer of Sketch the graph of the convex hull...

Question 38

Find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.&#10;

Accepted Answer

The answer of Find the barycentric coordinates of p with...

Question 39

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 40

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 41

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 42

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 43

Provide an appropriate response&#10;Which of the following statements are true? I: If A and B are convex sets then A + B is convex.&#10;II: A four dimensional polytope always has the same number of vertices and edges.&#10;A)II&#10;B)Neither I or II&#10;C)Both I and II&#10;D)I

Accepted Answer

The answer of Provide an appropriate response&#10;Which of the following...

Question 44

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 45

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 46

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 47

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 48

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 49

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 50

Provide an appropriate response&#10;Let int S be the set of all interior points of S, and let cl S be the closure of S (S &#8746; the set of all boundary points of S). Which of the following statements are true?&#10;I: If S is convex, then int S is convex.&#10;II: If S is convex, then cl S is convex.&#10;A)II&#10;B)Both I and II&#10;C)Neither I or II&#10;D)I

Accepted Answer

The answer of Provide an appropriate response&#10;Let int S be...

Question 51

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 52

Provide an appropriate response&#10;Which of the following statements are true? I: A polytope is the affine hull of a finite set of points.&#10;II: An extreme point of a polytope P is any point in the convex hull of 2 vertices.&#10;A)Neither I or II&#10;B)II&#10;C)I&#10;D)Both I and II

Accepted Answer

The answer of Provide an appropriate response&#10;Which of the following...

Question 53

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 54

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 55

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 56

Provide an appropriate response&#10;

Accepted Answer

The answer of Provide an appropriate response&#10; ...

Question 57

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Deck 8: The Geometry of Vector Spaces