Find a list of four distinct vectors in 𝐅^3 whose span equals
{(𝑥, 𝑦, 𝑧) ∈ 𝐅^3
∶ 𝑥 + 𝑦 + 𝑧 = 0}.

To find a list of four distinct vectors in \( \mathbb{F}^3 \) whose span equals the subspace \( \{(x, y, z) \in \mathbb{F}^3 : x + y + z = 0\} \), let's break down the problem step by step.

### 1. **Identify the Subspace:**
   The subspace \( S = \{(x, y, z) \in \mathbb{F}^3 : x + y + z = 0\} \) consists of all vectors in \( \mathbb{F}^3 \) where the sum of the components is zero. This is a plane in \( \mathbb{F}^3 \), meaning it's a two-dimensional subspace of \( \mathbb{F}^3 \).

### 2. **Determine the Dimension:**
   The equation \( x + y + z = 0 \) imposes one linear constraint on the vectors in \( \mathbb{F}^3 \), so the dimension of this subspace is \( 3 - 1 = 2 \).

### 3. **Find a Basis:**
   To find vectors that span this subspace, we need to find two linearly independent vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) that satisfy the equation \( x + y + z = 0 \). These vectors will form a basis for the subspace.

   Two possible vectors could be:
   \[
   \mathbf{v}_1 = (1, -1, 0), \quad \mathbf{v}_2 = (0, 1, -1)
   \]

   These vectors are linearly independent, and any vector in the subspace can be written as a linear combination of these two vectors.

### 4. **Find Four Distinct Vectors:**
   Although the dimension of the subspace is 2, we can construct four distinct vectors by choosing appropriate linear combinations. For instance:
   \[
   \mathbf{v}_1 = (1, -1, 0)
   \]
   \[
   \mathbf{v}_2 = (0, 1, -1)
   \]
   \[
   \mathbf{v}_3 = (-1, 0, 1) = -\mathbf{v}_1 + \mathbf{v}_2
   \]
   \[
   \mathbf{v}_4 = (1, 1, -2) = \mathbf{v}_1 + \mathbf{v}_2
   \]

### 5. **Verification:**
   The set of vectors \( \{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\} \) spans the entire subspace since any vector \( \mathbf{v} \) in the subspace can be written as a linear combination of \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \), and therefore also in terms of \( \mathbf{v}_3 \) and \( \mathbf{v}_4 \).

### **Final List of Vectors:**
- \( \mathbf{v}_1 = (1, -1, 0) \)
- \( \mathbf{v}_2 = (0, 1, -1) \)
- \( \mathbf{v}_3 = (-1, 0, 1) \)
- \( \mathbf{v}_4 = (1, 1, -2) \)

These four distinct vectors span the subspace \( \{(x, y, z) \in \mathbb{F}^3 : x + y + z = 0\} \).

Would you like any details or have any questions?

Here are 5 related questions:
1. How can we verify if a set of vectors spans a subspace?
2. What does it mean for vectors to be linearly independent?
3. How do we determine the dimension of a subspace?
4. What is the geometric interpretation of a plane in \( \mathbb{F}^3 \)?
5. How can we find the orthogonal complement of a subspace?

**Tip:** When finding a spanning set for a subspace, it's helpful to start by determining the dimension of the subspace and finding a basis, which is the smallest set of vectors needed to span the space.

Discover how to find four distinct vectors in 𝐅^3 that span the subspace defined by the equation x + y + z = 0. This problem explores fundamental concepts in linear algebra, including vector spaces, subspaces, and basis construction. Ideal for advanced undergraduate students and anyone interested in linear algebra applications.

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