https://file.aiquickdraw.com/mathbot/file/1724569914430zykugxd4.jpg

Choose A, B or C.
After that, solve this:
Find specific vectors u and v in W such that uplusv is not in W. This is enough to show that W is not a vector space.
Two vectors in​ W, u and v​, for which u plus v is not in W are

To address the problems in the image, I'll start by analyzing the multiple-choice question provided.

### Multiple-Choice Question Analysis

The question appears to be about determining whether a given vector belongs to a set \( W \) or not. Let's break it down:

- **Given:** The set \( W \) is the union of two sets. The vector in question is:
  \[
  \begin{pmatrix} x \\ y \end{pmatrix}
  \]
- **Choices:**
  - **Option A:** \( \begin{pmatrix} x \\ y \end{pmatrix} \) is in \( W \) then this vector could be \( \begin{pmatrix} x \\ cx \end{pmatrix} \) or \( \begin{pmatrix} cy \\ y \end{pmatrix} \) because \( cy = y \) or \( x = cx \) implies \( c = 1 \).
  - **Option B:** \( \begin{pmatrix} x \\ y \end{pmatrix} \) is in \( W \) then this vector could be \( \begin{pmatrix} x \\ cx \end{pmatrix} \), but not in \( W \) because only \( c = 0 \) in some cases.
  - **Option C:** \( \begin{pmatrix} x \\ y \end{pmatrix} \) is in \( W \) then this vector could be \( \begin{pmatrix} x \\ cx \end{pmatrix} \) or \( \begin{pmatrix} cy \\ y \end{pmatrix} \) but not in \( W \) because \( cy = 0 \) or \( x = y = 0 \).

### Step 1: Choosing the Correct Option

The key here is to determine under which conditions the vector \( \begin{pmatrix} x \\ y \end{pmatrix} \) would remain in \( W \). Let's interpret the options carefully:

- **Option A** seems to claim that the vector could be in \( W \) if certain conditions hold true (like \( c = 1 \)).
- **Option B** suggests that the vector would not always be in \( W \) because \( c \) might only be valid under certain conditions.
- **Option C** states that the vector could be in \( W \) but also indicates scenarios where it might not be in \( W \) due to constraints like \( cy = 0 \).

Since **Option B** appears to provide a valid restriction for the vector to not be in \( W \), **Option B** seems to be the correct choice.

### Step 2: Finding Specific Vectors \( u \) and \( v \) in \( W \)

We need to find two vectors \( u \) and \( v \) in \( W \) such that their sum \( u + v \) is not in \( W \). 

Given the structure of \( W \) and the union of sets, you can consider vectors that lie on different parts of the union, which when added do not satisfy the conditions for either part.

- Consider \( u = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) and \( v = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \).
  - Both \( u \) and \( v \) could be in \( W \) if \( W \) includes vectors aligned with the x-axis and y-axis, respectively.
  - However, their sum \( u + v = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \) may not lie in \( W \), depending on how \( W \) is defined. If \( W \) requires vectors to have a specific form (like \( cy = 0 \) or \( cx = 0 \)), then \( \begin{pmatrix} 1 \\ 1 \end{pmatrix} \) might not satisfy this.

Thus, \( u = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) and \( v = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \) are two vectors in \( W \) whose sum is not in \( W \).

This shows that \( W \) is not a vector space because it fails the closure property under vector addition.

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### Do you want more details or have any questions?

### Related Questions:
1. What are the key properties of a vector space?
2. How can you determine whether a subset of a vector space is also a vector space?
3. What are the implications of a set not being closed under vector addition?
4. Can you give another example of vectors \( u \) and \( v \) where \( u+v \) is not in \( W \)?
5. How does the choice of \( c \) affect the membership of vectors in \( W \)?

### Tip:
Always verify that both vector addition and scalar multiplication hold within a set to determine if it's a vector space.

Explore vector spaces and the closure property in mathematics with detailed examples. Learn how to identify specific vectors u and v within a vector space W such that their sum u + v is not in W, demonstrating that W is not a vector space. Suitable for advanced undergraduate students and those interested in algebraic structures.

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