Read the following set of vectors:

• v₁ = [1, 2, 3]

• v₂ = [2, 4, 6]

• v₃ = [3, 6, 9]

1. Determine whether the set {v₁, v₂, v₃} is linearly independent or dependent.

2. Justify your answer using the definition of linear dependence.

3. Reorganize the vectors and write any vector as a linear combination of the others (if applicable).

Extend this logic: Can a set of 4 vectors in R³ ever be linearly independent? Why or why not?

Task 2: Python Implementation

Write a Python function using NumPy to check whether a given set of vectors is linearly independent or not. Use matrix rank for your logic.

Instructions:

1. Define three vectors:

v1 = np.array([1, 2, 3])  

v2 = np.array([2, 4, 6])  

v3 = np.array([1, 0, 1])  

1. Stack the vectors into a matrix.

2. Use np.linalg.matrix_rank() to check independence.

Return whether the set is independent or dependent.

What type of vector must always be present in a subspace of a vector space?