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.

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.