To determine whether the vectors $\mathbf{X_1} = [2, 1, 1]$, $\mathbf{X_2} = [1, 3, 1]$, and $\mathbf{X_3} = [1, 2, -1]$ are linearly independent or dependent, we can use the following approach:

Step 1: Set up the equation for linear dependence

The vectors $\mathbf{X_1}, \mathbf{X_2}, \mathbf{X_3}$ are linearly dependent if there exist scalars $c_1, c_2, c_3$, not all zero, such that:

$c_1 \mathbf{X_1} + c_2 \mathbf{X_2} + c_3 \mathbf{X_3} = 0$

This gives the system of equations:

[ c_1 [2, 1, 1] + c_2 [1, 3, 1] + c_3 [1, 2, -1] = [0, 0, 0] ]

This can be written as the following set of equations:

2c_1 + c_2 + c_3 = 0 \tag{1} c_1 + 3c_2 + 2c_3 = 0 \tag{2} c_1 + c_2 - c_3 = 0 \tag{3}

Step 2: Solve the system of equations

We solve this system of three linear equations to see if the only solution is $c_1 = c_2 = c_3 = 0$. If it is, then the vectors are linearly independent. If there are non-zero solutions, the vectors are linearly dependent.

We will solve the system using substitution or elimination.

From Equation (3):

c_1 + c_2 - c_3 = 0 \implies c_1 = c_3 - c_2 \tag{4}

Substitute $c_1 = c_3 - c_2$ into Equations (1) and (2):

• Equation (1): $2(c_3 - c_2) + c_2 + c_3 = 0$ $2c_3 - 2c_2 + c_2 + c_3 = 0$ 3c_3 - c_2 = 0 \implies c_2 = 3c_3 \tag{5}

• Equation (2): $(c_3 - c_2) + 3c_2 + 2c_3 = 0$ $c_3 - c_2 + 3c_2 + 2c_3 = 0$ $3c_3 + 2c_2 = 0$ Substitute $c_2 = 3c_3$ into this equation: $3c_3 + 2(3c_3) = 0$ $3c_3 + 6c_3 = 0$ $9c_3 = 0 \implies c_3 = 0$

If $c_3 = 0$, then from Equation (5):

$c_2 = 3c_3 = 0$

Finally, substitute $c_2 = 0$ and $c_3 = 0$ into Equation (4):

$c_1 = c_3 - c_2 = 0$

Step 3: Conclusion

The only solution to this system is $c_1 = c_2 = c_3 = 0$. Therefore, the vectors $\mathbf{X_1}, \mathbf{X_2}, \mathbf{X_3}$ are linearly independent.

Would you like to explore the geometric interpretation or see related problems?

Here are some related questions you could consider:

1. What happens if the vectors were in $\mathbb{R}^4$ instead of $\mathbb{R}^3$?
2. How can you use the determinant of a matrix formed by vectors to check for linear independence?
3. How does the concept of linear dependence apply to solving systems of linear equations?
4. What is the rank of a matrix formed by the vectors $X_1, X_2, X_3$?
5. How can you generalize the method for $n$-dimensional space?

Tip: To quickly test linear dependence in higher dimensions, consider creating the matrix formed by the vectors and calculating its determinant. If the determinant is zero, the vectors are dependent; if non-zero, they are independent.