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is_linearly_independent

Checks if a set of vectors are linearly independent.

is_linearly_independent 

is_linearly_independent(vectors: list[ndarray]) -> bool

Check if set of vectors are linearly independent 1.

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.

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]))

True

Parameters:

  • vectors (list[ndarray]) –

    Vectors to check the linear independence of.

Returns:

  • bool –

    Return True if vectors are linearly independent False otherwise.

References

1 Wikipedia. Linear independence. link.

Source code in toqito/matrix_props/is_linearly_independent.py 
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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))