To determine the value of $c$ for which the set $S = \{(1, 2, -1), (3, 1, 0), (-1, 2, c)\}$ is linearly dependent, we need to check the condition for linear dependence.

Linear Dependence Condition:

Three vectors $\mathbf{v}_1 = (1, 2, -1)$, $\mathbf{v}_2 = (3, 1, 0)$, and $\mathbf{v}_3 = (-1, 2, c)$ are linearly dependent if there exist scalars $\alpha_1, \alpha_2, \alpha_3$, not all zero, such that:

$\alpha_1 \mathbf{v}_1 + \alpha_2 \mathbf{v}_2 + \alpha_3 \mathbf{v}_3 = 0$

This can be rewritten as a system of linear equations:

$\alpha_1 \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \alpha_2 \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix} + \alpha_3 \begin{pmatrix} -1 \\ 2 \\ c \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$

This results in the following system of equations:

1. $\alpha_1 + 3\alpha_2 - \alpha_3 = 0$
2. $2\alpha_1 + \alpha_2 + 2\alpha_3 = 0$
3. $-\alpha_1 + c\alpha_3 = 0$

For the vectors to be linearly dependent, the determinant of the matrix formed by the vectors must be zero. This matrix is:

1 & 3 & -1 \\ 2 & 1 & 2 \\ -1 & 0 & c \end{pmatrix}$$ We calculate the determinant of $$A$$: $$\text{det}(A) = 1 \cdot \begin{vmatrix} 1 & 2 \\ 0 & c \end{vmatrix} - 3 \cdot \begin{vmatrix} 2 & 2 \\ -1 & c \end{vmatrix} + (-1) \cdot \begin{vmatrix} 2 & 1 \\ -1 & 0 \end{vmatrix}$$ Now, compute each of these 2x2 determinants: $$\begin{vmatrix} 1 & 2 \\ 0 & c \end{vmatrix} = 1 \cdot c - 2 \cdot 0 = c$$ $$\begin{vmatrix} 2 & 2 \\ -1 & c \end{vmatrix} = 2c - (-2) = 2c + 2$$ $$\begin{vmatrix} 2 & 1 \\ -1 & 0 \end{vmatrix} = 2 \cdot 0 - 1 \cdot (-1) = 1$$ Substituting these values back into the determinant formula: $$\text{det}(A) = 1 \cdot c - 3 \cdot (2c + 2) - 1 \cdot 1$$ Simplifying: $$\text{det}(A) = c - 3(2c + 2) - 1 = c - 6c - 6 - 1 = -5c - 7$$ For the vectors to be linearly dependent, we need: $$\text{det}(A) = 0$$ Solving for $$c$$: $$-5c - 7 = 0$$ $$-5c = 7$$ $$c = -\frac{7}{5}$$ ### Conclusion: The value of $$c$$ for which the set $$S = \{(1, 2, -1), (3, 1, 0), (-1, 2, c)\}$$ is linearly dependent is $$c = -\frac{7}{5}$$. --- Would you like further explanation or have any questions? Here are some follow-up questions: 1. How can we interpret linear dependence geometrically in $$\mathbb{R}^3$$? 2. How does changing the value of $$c$$ affect the vectors' span? 3. What happens if you add or subtract a scalar multiple of one vector from another in a set of vectors? 4. What does it mean for a set of vectors to span $$\mathbb{R}^3$$? 5. How can you determine if a set of vectors is linearly independent? **Tip:** To verify whether vectors are linearly independent or dependent, always check the determinant of the matrix formed by placing the vectors as columns in the matrix. If the determinant is zero, they are dependent.