Check whether the following set of vectors is LD or LI ?
(a) S1 = {(1, 0, 0),(−2, 1, 0),(0, 1, 1)}

To determine whether the set of vectors \( S_1 = \{(1, 0, 0), (-2, 1, 0), (0, 1, 1)\} \) is linearly dependent (LD) or linearly independent (LI), we need to check if there exist scalars \( c_1, c_2, c_3 \), not all zero, such that:

\[
c_1(1, 0, 0) + c_2(-2, 1, 0) + c_3(0, 1, 1) = (0, 0, 0)
\]

This gives the following system of linear equations:

1. \( c_1 - 2c_2 = 0 \) (from the \( x \)-component)
2. \( c_2 + c_3 = 0 \) (from the \( y \)-component)
3. \( c_3 = 0 \) (from the \( z \)-component)

### Step 1: Solve the system of equations

- From equation (3), we have \( c_3 = 0 \).
- Substitute \( c_3 = 0 \) into equation (2), giving \( c_2 = 0 \).
- Substitute \( c_2 = 0 \) into equation (1), giving \( c_1 = 0 \).

### Step 2: Conclusion

Since the only solution to this system is \( c_1 = 0 \), \( c_2 = 0 \), and \( c_3 = 0 \), the set of vectors is **linearly independent (LI)**.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:
1. What does it mean for a set of vectors to be linearly independent?
2. How would the process change if the set were in four dimensions?
3. What is the significance of a linearly dependent set of vectors in vector spaces?
4. Can you check the independence of a set of more than three vectors using the same method?
5. How does this method apply to vectors in a general \( n \)-dimensional space?

**Tip**: For a quick check of linear independence, put the vectors as rows in a matrix and calculate the determinant—if it's non-zero, the vectors are independent.

Check whether the following set of vectors is LD or LI? (a) S1 = {(1, 0, 0), (−2, 1, 0), (0, 1, 1)}

Learn how to determine if the set of vectors S1 = {(1, 0, 0), (-2, 1, 0), (0, 1, 1)} is linearly independent. Follow a detailed step-by-step solution, including solving a system of linear equations and applying the Linear Independence Theorem. Suitable for undergraduate students studying Linear Algebra.

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