### Instruction: The set of vectors $\left\{ \begin{pmatrix} 3 \\ 7 \end{pmatrix}, \begin{pmatrix} k \\ -2 \end{pmatrix} \right\}$ is linearly independent.  Find all possible values of $k.$  Enter your answer as an interval.### Response: I know that two vectors are linearly independent if and only if they are not parallel, which means that their slopes are not equal.
So I need to find the values of $k$ that make the slope of the second vector different from the slope of the first vector.
The slope of the first vector is $\frac{7}{3}$ and the slope of the second vector is $\frac{-2}{k}$.
I want to avoid the situation where $\frac{7}{3} = \frac{-2}{k}$, which would make the vectors parallel and linearly dependent.
So I cross-multiply and get $7k = -6$.
This implies that $k = -\frac{6}{7}$ is the only value of $k$ that makes the vectors parallel and linearly dependent.
Therefore, any other value of $k$ would make the vectors linearly independent.