Find a list of four distinct vectors in 𝐅3 whose span equals {(𝑥,𝑦,𝑧)∈𝐅3 ∶𝑥+𝑦+𝑧=0}.

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.

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

Explore how to find four distinct vectors in F^3 that span the subspace defined by x + y + z = 0. This problem involves understanding vector spaces, linear combinations, and the concept of span in linear algebra. Suitable for undergraduate students studying linear algebra.

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