If C is the field of complex numbers, which vectors in C
3
are linear combinations of (1, 0, −1), (0, 1, 1), and
(1, 1, 1)?

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 \((1, 1, 1)\).

---

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.

If C is the field of complex numbers, which vectors in C^3 are linear combinations of (1, 0, −1), (0, 1, 1), and (1, 1, 1)?

Learn how to determine which vectors in C^3 can be expressed as linear combinations of the vectors (1, 0, -1), (0, 1, 1), and (1, 1, 1). This detailed solution covers key concepts in linear algebra, including systems of linear equations, complex numbers, and spanning sets. Suitable for students in Grades 11-12.

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