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.