Source code for toqito.matrix_props.is_linearly_independent

"""Checks if a set of vectors are linearly independent."""

import numpy as np




def is_linearly_independent(vectors: list[np.ndarray]) -> bool:
    r"""Check if set of vectors are linearly independent [@WikiLinearIndependence].

    Examples:
        The following vectors are an example of a linearly independent set of vectors in \(\mathbb{R}^3\).

        \[
            \begin{pmatrix}
                1 \\ 0 \\ 1
            \end{pmatrix}, \quad
            \begin{pmatrix}
                1 \\ 1 \\ 0
            \end{pmatrix}, \quad \text{and} \quad
            \begin{pmatrix}
                0 \\ 0 \\ 1
            \end{pmatrix}
        \]

        We can see that these are linearly independent.

        ```python exec="1" source="above"
        import numpy as np
        from toqito.matrix_props import is_linearly_independent

        v_1 = np.array([[1], [0], [1]])
        v_2 = np.array([[1], [1], [0]])
        v_3 = np.array([[0], [0], [1]])

        print(is_linearly_independent([v_1, v_2, v_3]))
        ```

    Args:
        vectors: Vectors to check the linear independence of.

    Returns:
        Return `True` if vectors are linearly independent `False` otherwise.

    """
    # Check if the rank of the matrix equals the number of vectors.
    return bool(np.linalg.matrix_rank(np.column_stack(vectors)) == len(vectors))