We are asked to find which vectors in $\mathbb{C}^3$ (the space of 3-dimensional complex vectors) can be written as linear combinations of the three given vectors: $(1, 0, -1)$, $(0, 1, 1)$, and $(1, 1, 1)$.

A vector $\mathbf{v} = (x, y, z) \in \mathbb{C}^3$ is a linear combination of $(1, 0, -1)$, $(0, 1, 1)$, and $(1, 1, 1)$ if there exist complex scalars $a$, $b$, and $c$ such that:

$a(1, 0, -1) + b(0, 1, 1) + c(1, 1, 1) = (x, y, z)$

This results in the following system of equations for the components of the vectors:

$a + c = x \quad \text{(from the first component)}$ $b + c = y \quad \text{(from the second component)}$ $-a + b + c = z \quad \text{(from the third component)}$

Now, let's solve this system for $a$, $b$, and $c$ in terms of $x$, $y$, and $z$.

Step 1: Solve the system

1. From the first equation $a + c = x$, we can solve for $a$:

$a = x - c$

2. From the second equation $b + c = y$, solve for $b$:

$b = y - c$

3. Substitute $a = x - c$ and $b = y - c$ into the third equation $-a + b + c = z$:

$-(x - c) + (y - c) + c = z$ $-x + c + y - c + c = z$ $-x + y + c = z$

4. Solve for $c$:

$c = z + x - y$

Step 2: Find $a$ and $b$

Now that we know $c = z + x - y$, substitute this back into the expressions for $a$ and $b$:

$a = x - (z + x - y) = y - z$ $b = y - (z + x - y) = 2y - x - z$

Step 3: Conclusion

Thus, for any vector $\mathbf{v} = (x, y, z) \in \mathbb{C}^3$, we can write it as a linear combination of the given vectors if and only if $a = y - z$, $b = 2y - x - z$, and $c = z + x - y$, where $a$, $b$, and $c$ are complex scalars.

This means that every vector in $\mathbb{C}^3$ can be expressed as a linear combination of the given three vectors, and they span $\mathbb{C}^3$. Therefore, any vector in $\mathbb{C}^3$ is a linear combination of $(1, 0, -1)$, $(0, 1, 1)$, and $(1, 1, 1)$.

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Would you like further details on this process or clarification on any step? Here are some related questions that might help:

1. How do we check if a set of vectors is linearly independent?
2. What is the concept of a vector space basis, and how does it relate to this problem?
3. How can we extend this idea to higher-dimensional spaces like $\mathbb{C}^n$?
4. What is the rank of a matrix, and how does it relate to the span of vectors?
5. How does the solution change if we work with real numbers instead of complex numbers?

Tip: When working with linear combinations, it's helpful to start by writing out the equations clearly, then use substitution or elimination to solve for the coefficients of the linear combination.