To determine whether the given set of vectors $\mathbf{X_1} = [1, 2, -1, 3]$, $\mathbf{X_2} = [2, -1, 3, 2]$, and $\mathbf{X_3} = [-1, 8, -9, 5]$ are linearly dependent, we must solve the equation:

$c_1\mathbf{X_1} + c_2\mathbf{X_2} + c_3\mathbf{X_3} = 0$

This expands into:

$c_1 \begin{bmatrix} 1 \\ 2 \\ -1 \\ 3 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ -1 \\ 3 \\ 2 \end{bmatrix} + c_3 \begin{bmatrix} -1 \\ 8 \\ -9 \\ 5 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$

Which corresponds to the system of equations:

1. & \; c_1 + 2c_2 - c_3 = 0 \\ 2. & \; 2c_1 - c_2 + 8c_3 = 0 \\ 3. & \; -c_1 + 3c_2 - 9c_3 = 0 \\ 4. & \; 3c_1 + 2c_2 + 5c_3 = 0 \end{aligned}$$ We solve this system by forming the augmented matrix and reducing it. Let me calculate this for you.import numpy as np # Define the coefficient matrix and the augmented matrix A = np.array([ [1, 2, -1], [2, -1, 8], [-1, 3, -9], [3, 2, 5] ]) # Augmented with a zero column for the homogeneous system b = np.zeros(4) # Perform row-reduction or solve for null space to check dependence rank_A = np.linalg.matrix_rank(A) null_space_dim = A.shape[1] - rank_A # Compute null space using SVD for relations if linearly dependent u, s, vh = np.linalg.svd(A) null_space = vh.T[:, rank_A:] null_space_dim, null_space(1, array([[-0.80178373], [ 0.53452248], [ 0.26726124]]))(1, array([[-0.80178373], [ 0.53452248], [ 0.26726124]]))successThe given set of vectors is **linearly dependent**, as the null space dimension is 1. The relationship between the vectors can be expressed as: $$-0.80178373 \cdot \mathbf{X_1} + 0.53452248 \cdot \mathbf{X_2} + 0.26726124 \cdot \mathbf{X_3} = 0$$ To simplify, we normalize the coefficients by dividing by $$0.26726124$$ (the smallest non-zero value): $$-3 \cdot \mathbf{X_1} + 2 \cdot \mathbf{X_2} + 1 \cdot \mathbf{X_3} = 0$$ Thus, the relation is: $$3\mathbf{X_1} = 2\mathbf{X_2} + \mathbf{X_3}$$ This confirms that the vectors are linearly dependent. Do you want further details or calculations? Here are 5 related questions for exploration: 1. How can the dependency relationship be visually interpreted in 4D space? 2. What happens if another vector is added to the set? Would it still be dependent? 3. Can we apply this method to vectors of arbitrary dimensions? 4. How does the rank of the matrix determine linear independence? 5. What are real-world applications of detecting linear dependence among vectors? **Tip:** Linearly dependent vectors can always be expressed as a combination of others in the set!