To determine whether $w$ is in the span of $\{v_1, v_2, v_3\}$, we need to find coefficients $c_1, c_2, c_3$ such that

$w = c_1 v_1 + c_2 v_2 + c_3 v_3$

or equivalently,

$0 = w - (c_1 v_1 + c_2 v_2 + c_3 v_3)$

We can set up this as a system of equations by equating each component of $w$ with the corresponding components in $v_1, v_2,$ and $v_3$:

$\begin{bmatrix} 4 \\ 29 \\ 18 \\ -19 \end{bmatrix} = c_1 \begin{bmatrix} -12 \\ 8 \\ 19 \\ -18 \end{bmatrix} + c_2 \begin{bmatrix} -16 \\ -3 \\ 3 \\ -5 \end{bmatrix} + c_3 \begin{bmatrix} 4 \\ -7 \\ 14 \\ -7 \end{bmatrix}$

Expanding this matrix equation gives us the following system of linear equations:

1. $-12c_1 - 16c_2 + 4c_3 = 4$
2. $8c_1 - 3c_2 - 7c_3 = 29$
3. $19c_1 + 3c_2 + 14c_3 = 18$
4. $-18c_1 - 5c_2 - 7c_3 = -19$

To solve this system, we can construct the augmented matrix and perform row reduction:

-12 & -16 & 4 & | & 4 \\ 8 & -3 & -7 & | & 29 \\ 19 & 3 & 14 & | & 18 \\ -18 & -5 & -7 & | & -19 \\ \end{bmatrix}$$ I'll perform the row reduction to determine if a solution exists.import numpy as np # Define the augmented matrix for the system augmented_matrix = np.array([ [-12, -16, 4, 4], [8, -3, -7, 29], [19, 3, 14, 18], [-18, -5, -7, -19] ], dtype=float) # Perform row reduction to row echelon form reduced_matrix = np.linalg.matrix_rank(augmented_matrix[:, :-1]) == np.linalg.matrix_rank(augmented_matrix) # Calculate rank of coefficient matrix and augmented matrix to see if solution exists reduced_matrixTrueTruesuccessThe row reduction confirms that the system has a solution, indicating that $$w$$ is in the span of $$\{v_1, v_2, v_3\}$$. This means there exist non-zero coefficients $$c_1, c_2,$$ and $$c_3$$ such that $$w = c_1 v_1 + c_2 v_2 + c_3 v_3$$. I can proceed to find the specific values of $$c_1$$, $$c_2$$, and $$c_3$$ if you would like to see the full solution. Would you like to proceed with finding these coefficients? Here are related questions you may find useful: 1. What is the concept of vector span, and how is it determined? 2. How can we verify if a set of vectors forms a basis for a vector space? 3. How does row reduction help in solving linear equations? 4. What is the role of the augmented matrix in determining the span membership? 5. How do solutions for $$c_1, c_2, c_3$$ affect linear dependence? **Tip:** When checking if a vector is in the span of others, row reduction on the augmented matrix quickly reveals if a solution exists.