We need to find four distinct vectors in $\mathbb{F}^3$ whose span forms the set $\{(x, y, z) \in \mathbb{F}^3 : x + y + z = 0\}$.

Step 1: Define the Subspace

The subspace is defined by the equation $x + y + z = 0$. This is a plane through the origin in $\mathbb{F}^3$, meaning the subspace has dimension 2 (since it's a plane, it will have two independent directions, or two basis vectors).

Step 2: Find Basis for the Subspace

To find a basis for this subspace, we can express $z$ in terms of $x$ and $y$, as follows:

$z = -(x + y)$

This means each vector in the subspace has the form:

$(x, y, -(x + y)) = x(1, 0, -1) + y(0, 1, -1)$

Thus, the two basis vectors for the subspace are:

$v_1 = (1, 0, -1)$ $v_2 = (0, 1, -1)$

Step 3: Form Four Distinct Vectors in the Subspace

Any vector in this subspace is a linear combination of $v_1$ and $v_2$. We can select different linear combinations to generate four distinct vectors:

1. $v_1 = (1, 0, -1)$
2. $v_2 = (0, 1, -1)$
3. $v_3 = v_1 + v_2 = (1, 1, -2)$
4. $v_4 = v_1 - v_2 = (1, -1, 0)$

Step 4: Check the Span

The span of these vectors is the set of all linear combinations of $v_1$ and $v_2$, which gives us the full subspace $\{(x, y, z) \in \mathbb{F}^3 : x + y + z = 0\}$, as required.

Conclusion

Thus, the four distinct vectors that span this subspace are:

$\{(1, 0, -1), (0, 1, -1), (1, 1, -2), (1, -1, 0)\}$

Would you like any further details or clarifications on this?

Related Questions:

1. What is the dimension of the vector space $\mathbb{F}^3$?
2. How do you determine if a set of vectors is linearly independent?
3. Can a subspace of $\mathbb{F}^3$ have a dimension greater than 2?
4. What are the properties of subspaces in finite-dimensional vector spaces?
5. How do you generalize the span of vectors to higher dimensions?

Tip:

When finding a basis for a subspace, it helps to express one variable in terms of the others to reduce the number of free variables.