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The image contains a problem related to linear algebra, specifically dealing with vector spaces and linear independence. Here's a translation and breakdown of the problem:

### Problem Statement:
You have a vector space \( V \) defined as \( V = \mathbb{R}^3 \) with the basis vectors \( \{(1,0,0), (0,1,0), (0,0,1)\} \). You are given a set of vectors \( \{ \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_4} \} \) where:

- \( \mathbf{v_1} = (1,2,3) \)
- \( \mathbf{v_2} = (4,5,6) \)
- \( \mathbf{v_3} = (7,8,9) \)
- \( \mathbf{v_4} = (1,1,1) \)

The problem asks for the following:

1. **Determine if the vectors \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_4} \) are linearly independent.**
2. **Find the dimension of the span of \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_4} \).**
3. **Find a basis for the span of \( \{ \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_4} \} \).**

### Step 1: Check Linear Independence

To check if the vectors are linearly independent, we need to check if the only solution to the equation

\[
c_1\mathbf{v_1} + c_2\mathbf{v_2} + c_3\mathbf{v_3} + c_4\mathbf{v_4} = \mathbf{0}
\]

is the trivial solution \( c_1 = c_2 = c_3 = c_4 = 0 \).

This corresponds to solving the following system of linear equations:

\[
\begin{pmatrix}
1 & 4 & 7 & 1 \\
2 & 5 & 8 & 1 \\
3 & 6 & 9 & 1
\end{pmatrix}
\begin{pmatrix}
c_1 \\
c_2 \\
c_3 \\
c_4
\end{pmatrix}
=
\begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}
\]

### Step 2: Find the Dimension of the Span

The dimension of the span is the rank of the matrix formed by these vectors. The rank is the maximum number of linearly independent rows or columns in the matrix.

### Step 3: Find a Basis for the Span

To find a basis, we determine which of the vectors are linearly independent. The independent vectors will form the basis.

---

I'll calculate these steps for you now.It seems there was an issue with the method I initially used. I'll recalculate the rank using a different approach and check the linear independence.### Results:

1. **Linear Independence**:  
   The rank of the matrix formed by the vectors \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_4} \) is **2**. Since the rank (2) is less than the number of vectors (4), the vectors are **linearly dependent**.

2. **Dimension of the Span**:  
   The dimension of the span of the vectors \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_4} \) is **2**.

3. **Basis for the Span**:  
   Since the dimension of the span is 2, we need to find 2 linearly independent vectors among the given ones. Based on the rank, vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are sufficient to form a basis for the span. Therefore, the basis is:

   \[
   \text{Basis} = \{ \mathbf{v_1}, \mathbf{v_2} \} = \{(1,2,3), (4,5,6)\}
   \]

Would you like to explore any of these results further or have any questions?

---

### Related Questions:
1. What does it mean for vectors to be linearly dependent or independent?
2. How do you calculate the rank of a matrix, and why is it important?
3. What is the significance of the dimension of a vector space?
4. How can you use the Gram-Schmidt process to find an orthogonal basis?
5. What are some applications of vector spaces and linear independence in real life?

### Tip:
When working with vector spaces, always remember that the number of vectors in a basis equals the dimension of the vector space, which is a key property used in determining linear independence and span.

Explore how to determine linear independence and find the dimension and basis of vector spaces using a detailed example. This problem involves vectors in \( \mathbb{R}^3 \) and covers key concepts in linear algebra, including matrix equation solving.

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