Provide an Appropriate Response
- And $2 \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 } - 2 \mathbf { v } _ { 3 } - \mathbf { v } _ { 4 } = \mathbf { 0 }$

question 39

Multiple Choice

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.

• A) \(p = \frac { 5 } { 8 } \mathbf { v } _ { 1 } + \frac { 3 } { 16 } \mathbf { v } _ { 2 } + \frac { 3 } { 16 } \mathbf { v } _ { 4 }\)
• B) \(\mathbf { p } = \frac { 1 } { 4 } \mathbf { v } _ { 1 } + \frac { 3 } { 8 } \mathbf { v } _ { 2 } + \frac { 3 } { 8 } \mathbf { v } _ { 4 }\)
• C) \(\mathrm { p } = \frac { 1 } { 8 } \mathbf { v } _ { 2 } + \frac { 3 } { 8 } \mathbf { v } _ { 3 } + \frac { 1 } { 2 } \mathbf { v } _ { 4 }\)
• D) It cannot be done using only \(3 \mathbf { v } _ { \mathbf { i } }\) 's.

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Definitions:

Religious Bodies

Organized groups of individuals who share the same religious beliefs and practices, often structured with a hierarchy and common rituals.

Church Doctrine

The official teachings and beliefs of a religious institution, often based on sacred texts and traditions.

Breaking Away

The act of distancing oneself from previous affiliations or conventions, seeking independence or a new direction.

Sects

Usually form by breaking away from churches due to disagreement about church doctrine. Sects are less integrated into society and less bureaucratized than churches. They are often led by charismatic leaders, who tend to be relatively intolerant of religious opinions other than their own.