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To determine the span of the given set of vectors, we need to see if these vectors span the entire 3-dimensional space or just a plane within that space.

### Vectors:
\[
\mathbf{v_1} = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \mathbf{v_2} = \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}, \mathbf{v_3} = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}
\]

### Step-by-Step Solution:

1. **Construct a matrix using the vectors**: We will check the rank of the matrix formed by placing these vectors as columns:
\[
A = \begin{bmatrix} 
2 & 1 & 1 \\ 
1 & 0 & 0 \\ 
1 & 2 & 1 
\end{bmatrix}
\]

2. **Determine the rank**: The rank of a matrix tells us how many linearly independent vectors there are, which in turn tells us the dimension of the span of these vectors. We will perform row reduction (Gaussian elimination) to determine the rank.

Let's perform that calculation.

The rank of the matrix formed by the given vectors is 3. This means the vectors are linearly independent and span the entire 3-dimensional space.

### Conclusion:
The span of the vectors is **the entire 3-dimensional space**.

Would you like further details on the solution, or do you have any other questions?

### Related Questions:
1. What does it mean for vectors to span a space?
2. How is the rank of a matrix connected to the span of vectors?
3. What are linearly independent vectors?
4. How can you visually interpret the span of vectors in 2D and 3D spaces?
5. What methods are used to determine if a set of vectors spans a subspace?

### Tip:
The rank of a matrix is the maximum number of linearly independent rows or columns. In general, for vectors in \( \mathbb{R}^n \), if the rank equals \( n \), the vectors span the entire space.

What is the span of the following vectors: [2, 1, 1], [1, 0, 2], [1, 0, 1]?

Learn how to determine the span of the vectors [2, 1, 1], [1, 0, 2], and [1, 0, 1] using linear algebra concepts like matrix rank and linear independence. This problem is ideal for students studying linear algebra at an undergraduate level.

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