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.