Linear independence is a core linear algebra concept used to tell whether a set of vectors contains any redundancy. In this task, you’ll implement a simple checker that decides if a list of vectors is linearly independent.
Formally, a set of vectors $\{\mathbf{v}_1, \dots, \mathbf{v}_k\}$ is linearly independent if the equation:
$$
c_1 \mathbf{v}_1 + \dots + c_k \mathbf{v}_k = \mathbf{0}
$$
has only the trivial solution $c_1 = \dots = c_k = 0$.

## Requirements

Implement the function
```python
def is_linearly_independent(vectors):
    \"""
    Checks whether a set of vectors is linearly independent.

    Args:
        vectors (np.ndarray): A k x n array of k vectors, each of length n.

    Returns:
        bool: True if the vectors are linearly independent, otherwise False.
    \"""
    # Your solution below
```

**Rules:**
- Use the definition: vectors are linearly independent iff the only solution to $A\mathbf{c}=\mathbf{0}$ is $\mathbf{c}=\mathbf{0}$, where $A$ has the vectors as columns.
- Build matrix $A$ from the input list and determine independence via rank: independent iff $\text{rank}(A)=k$.
- Return a single boolean (`True`/`False`).
- Do not use symbolic math libraries or prebuilt “independence” helpers; rely on NumPy linear algebra (e.g., `np.linalg.matrix_rank`).

## Example

```python
vectors = np.array([
    [1.0, 0.0, 0.0],
    [0.0, 1.0, 0.0],
    [1.0, 1.0, 0.0]
])

is_linearly_independent(vectors)
```

Output:
```python
False
```

Stratified Experiment Estimator

Fix Broken Funnel Analysis

Linear independence

Solve this easy level ML coding question from General. The premier LeetCode for ML platform to master Python, SQL, and Machine Learning algorithms.

1194. Linear independence

Examples

Constraints

Similar Questions