To determine if the vectors $\mathbf{v_1} = (1, 3, 7)$, $\mathbf{v_2} = (1, 2, 0)$, and $\mathbf{v_3} = (1, 1, -7)$ are linearly independent, we need to check whether the only solution to the equation

$c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3} = \mathbf{0}$

is $c_1 = c_2 = c_3 = 0$. This can be done by forming a matrix with $\mathbf{v_1}$, $\mathbf{v_2}$, and $\mathbf{v_3}$ as columns and checking if the determinant is non-zero.

Step 1: Form the Matrix

1 & 1 & 1 \\ 3 & 2 & 1 \\ 7 & 0 & -7 \end{pmatrix}$$ ### Step 2: Calculate the Determinant $$\text{Det}(A) = 1 \cdot \begin{vmatrix} 2 & 1 \\ 0 & -7 \end{vmatrix} - 1 \cdot \begin{vmatrix} 3 & 1 \\ 7 & -7 \end{vmatrix} + 1 \cdot \begin{vmatrix} 3 & 2 \\ 7 & 0 \end{vmatrix}$$ $$\text{Det}(A) = 1 \cdot (2 \cdot -7 - 0 \cdot 1) - 1 \cdot (3 \cdot -7 - 1 \cdot 7) + 1 \cdot (3 \cdot 0 - 7 \cdot 2)$$ $$\text{Det}(A) = 1 \cdot (-14) - 1 \cdot (-21 - 7) + 1 \cdot (0 - 14)$$ $$\text{Det}(A) = -14 + 28 - 14 = 0$$ ### Step 3: Interpret the Result Since the determinant of the matrix $$A$$ is zero, this means the vectors $$\mathbf{v_1}$$, $$\mathbf{v_2}$$, and $$\mathbf{v_3}$$ are **not** linearly independent. Instead, they are linearly dependent. ### Final Answer - **False**: The vectors $$\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}$$ are not linearly independent. Would you like more details on linear dependence, or do you have any other questions? --- Here are five follow-up questions you might consider: 1. What does it mean for vectors to be linearly dependent? 2. How can you determine the rank of a matrix formed by vectors? 3. What is the significance of a zero determinant in the context of vector independence? 4. How can the Gram-Schmidt process be used to orthogonalize vectors? 5. What are some real-world applications where linear independence is crucial? **Tip:** When vectors are linearly dependent, at least one of the vectors can be expressed as a linear combination of the others.