To determine whether the given vectors are linearly independent, we can test if the equation

$c_1 \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} + c_2 \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} + c_3 \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + c_4 \begin{pmatrix} -3 \\ 7 \\ 2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$

has only the trivial solution, where $c_1 = c_2 = c_3 = c_4 = 0$.

Steps:

1. Set up the system of equations by forming a matrix with the vectors as columns and row reduce it.

   1 & 2 & 0 & -3 \\ 2 & -1 & 1 & 7 \\ 4 & 3 & 2 & 2 \end{pmatrix}$$

2. Apply Gaussian elimination or calculate the determinant if the matrix is square. Since this is a $3 \times 4$ matrix (more vectors than dimensions), we can row reduce it to check for linear dependence.

Let me solve this.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to explain how to solve this manually or go through the steps for row reduction?

Here are 5 related questions you might find interesting:

1. What is the geometric interpretation of linearly independent vectors?
2. How does Gaussian elimination help in determining linear independence?
3. Can linearly dependent vectors form a basis for a vector space?
4. What happens when the number of vectors exceeds the dimension of the space?
5. How do you calculate the rank of a matrix?

Tip: If a set of vectors is linearly dependent, at least one vector can be written as a linear combination of others.