To determine whether the vectors $\mathbf{v}_1 = [2, 1, 1]$, $\mathbf{v}_2 = [1, 3, 1]$, and $\mathbf{v}_3 = [1, 2, -1]$ are linearly dependent or independent, we check if there exist scalars $c_1, c_2, c_3$, not all zero, such that:

$c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3 \mathbf{v}_3 = 0.$

This expands to:

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

Breaking this into component 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 1: Write the system in matrix form

This system can be represented as:

2 & 1 & 1 \\ 1 & 3 & 2 \\ 1 & 1 & -1 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}.$$ ### Step 2: Solve the system We compute the determinant of the coefficient matrix to see if it is invertible. If the determinant is non-zero, the vectors are linearly independent; otherwise, they are dependent. $$\text{Determinant} = \begin{vmatrix} 2 & 1 & 1 \\ 1 & 3 & 2 \\ 1 & 1 & -1 \end{vmatrix}.$$ Using cofactor expansion along the first row: $$\text{Determinant} = 2 \begin{vmatrix} 3 & 2 \\ 1 & -1 \end{vmatrix} - 1 \begin{vmatrix} 1 & 2 \\ 1 & -1 \end{vmatrix} + 1 \begin{vmatrix} 1 & 3 \\ 1 & 1 \end{vmatrix}.$$ Calculate each minor: 1. $$\begin{vmatrix} 3 & 2 \\ 1 & -1 \end{vmatrix} = (3)(-1) - (2)(1) = -3 - 2 = -5$$, 2. $$\begin{vmatrix} 1 & 2 \\ 1 & -1 \end{vmatrix} = (1)(-1) - (2)(1) = -1 - 2 = -3$$, 3. $$\begin{vmatrix} 1 & 3 \\ 1 & 1 \end{vmatrix} = (1)(1) - (3)(1) = 1 - 3 = -2$$. Substitute back: $$\text{Determinant} = 2(-5) - 1(-3) + 1(-2) = -10 + 3 - 2 = -9.$$ ### Step 3: Interpret the result Since the determinant is $$-9 \neq 0$$, the coefficient matrix is invertible. This implies that the only solution to the system is the trivial solution ($$c_1 = c_2 = c_3 = 0$$). ### Conclusion: The vectors $$[2, 1, 1]$$, $$[1, 3, 1]$$, and $$[1, 2, -1]$$ are **linearly independent**. --- Would you like further clarification or additional steps explained? Here are some related questions to deepen understanding: 1. How do we compute determinants for larger matrices? 2. What does it mean geometrically for vectors to be linearly independent? 3. Can you provide an example of linearly dependent vectors in $$\mathbb{R}^3$$? 4. How does the rank of a matrix relate to linear independence? 5. What are the implications of a zero determinant in solving vector equations? **Tip**: Linear independence is closely related to the concept of basis vectors; if vectors are independent, they can span a space without redundancy.