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To determine which of the given sets of vectors in \(\mathbb{R}^2\) is linearly independent, we need to identify the set where the vectors do not form a linear combination that results in the zero vector unless all coefficients in that combination are zero.

### Analyzing Each Set

1. **Set A: \(\{(0, 1), (1, 0), (1, -1)\}\)**
   - This set contains three vectors in \(\mathbb{R}^2\). 
   - Since the dimension of \(\mathbb{R}^2\) is 2, any set of more than two vectors must be linearly dependent.

2. **Set B: \(\{(8, 0), (0, -5)\}\)**
   - This set contains two vectors.
   - Check if these vectors are linearly dependent:
     - The vectors are \((8, 0)\) and \((0, -5)\). These are not scalar multiples of each other, so they are linearly independent.

3. **Set C: \(\{(-1, 2), (1, -2)\}\)**
   - This set contains two vectors.
   - Check if these vectors are linearly dependent:
     - The vectors are \((-1, 2)\) and \((1, -2)\). These vectors are scalar multiples of each other (by multiplying \((-1, 2)\) by \(-1\), we get \((1, -2)\)). Therefore, they are linearly dependent.

4. **Set D: \(\{(1, 0), (-7, 0)\}\)**
   - This set contains two vectors.
   - Check if these vectors are linearly dependent:
     - The vectors are \((1, 0)\) and \((-7, 0)\). These vectors are scalar multiples of each other (by multiplying \((1, 0)\) by \(-7\), we get \((-7, 0)\)). Therefore, they are linearly dependent.

### Conclusion
- The only set that is linearly independent is **Set B: \(\{(8, 0), (0, -5)\}\)**. 

Would you like a more detailed explanation, or do you have any questions about this process?

### Related Questions:
1. What does it mean for vectors to be linearly dependent or independent?
2. How can we determine if more than two vectors are linearly independent in \(\mathbb{R}^n\)?
3. Can you explain the concept of the span of a set of vectors?
4. What is the significance of the number of vectors relative to the dimension of the space in determining linear independence?
5. How do you construct a basis for a vector space?

**Tip:** To test linear independence, set up a linear combination of the vectors equal to the zero vector and solve for the coefficients. If the only solution is the trivial solution (all coefficients are zero), the set is linearly independent.

Consider the following four sets of vectors A, B, C, D in R^2. Out of these sets of vectors, one set is linearly independent in R^2. Identify that linearly independent set.

Explore how to identify a linearly independent set of vectors in R^2 from given sets. Learn about linear independence, vector spaces, and key concepts in linear algebra. Suitable for undergraduate students in mathematics.

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