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To solve this question, we need to check the given conditions for the linear independence of the vectors \( \mathbf{x_1}, \mathbf{x_2}, \mathbf{x_3} \) and then verify which of the given options is consistent with that condition.

### Given Vectors:
- \( \mathbf{x_1} = (1, 1, -1, 1) \)
- \( \mathbf{x_2} = (1, -1, 2, -1) \)
- \( \mathbf{x_3} = (3, 1, 0, 1) \)

The question asks us to determine which equation holds if these vectors are **linearly independent**.

### Linear Independence Condition:
Vectors \( \mathbf{x_1}, \mathbf{x_2}, \mathbf{x_3} \) are linearly independent if the only solution to the equation \( c_1 \mathbf{x_1} + c_2 \mathbf{x_2} + c_3 \mathbf{x_3} = 0 \) is \( c_1 = c_2 = c_3 = 0 \).

Now, let's test each of the provided options:

#### a) \( 2 \mathbf{x_1} + \mathbf{x_2} = \mathbf{x_3} \)
- Left-hand side (LHS): \( 2(1, 1, -1, 1) + (1, -1, 2, -1) = (2, 2, -2, 2) + (1, -1, 2, -1) = (3, 1, 0, 1) \)
- Right-hand side (RHS): \( \mathbf{x_3} = (3, 1, 0, 1) \)

Since LHS = RHS, **option a** holds true.

#### b) \( \mathbf{x_1} + 2 \mathbf{x_2} = 3 \mathbf{x_3} \)
- LHS: \( (1, 1, -1, 1) + 2(1, -1, 2, -1) = (1, 1, -1, 1) + (2, -2, 4, -2) = (3, -1, 3, -1) \)
- RHS: \( 3(3, 1, 0, 1) = (9, 3, 0, 3) \)

LHS \( \neq \) RHS, so **option b** is false.

#### c) \( 3 (\mathbf{x_1} + \mathbf{x_2}) = \mathbf{x_3} \)
- LHS: \( 3((1, 1, -1, 1) + (1, -1, 2, -1)) = 3(2, 0, 1, 0) = (6, 0, 3, 0) \)
- RHS: \( \mathbf{x_3} = (3, 1, 0, 1) \)

LHS \( \neq \) RHS, so **option c** is false.

#### d) None of these
Since option a is true, **option d** is also false.

### Correct Answer:
The correct option is **a**.

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Would you like more details or have any questions on the steps? Here are five related questions for further understanding:

1. How do you determine if three vectors are linearly independent?
2. What is the geometric interpretation of linearly independent vectors?
3. Can two vectors be linearly independent in a 3-dimensional space?
4. What happens if vectors are linearly dependent? How would it affect this problem?
5. How does solving a system of linear equations relate to checking vector independence?

**Tip:** When testing vector equations, ensure that the left-hand side and right-hand side of the equation match in every component.

If the vectors x1 = (1, 1, -1, 1), x2 = (1, -1, 2, -1), and x3 = (3, 1, 0, 1) are linearly independent, then:

Solve the problem of checking the linear independence of vectors x1 = (1, 1, -1, 1), x2 = (1, -1, 2, -1), and x3 = (3, 1, 0, 1) by applying concepts of linear algebra. Explore the key theorem and conditions for vector independence.

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