Задача 2. Без базиса не выйдет

import numpy as np

class ShapeMismatchError(Exception):
    pass

def get_projections_components(
    matrix: np.ndarray,
    vector: np.ndarray,
):# -> tuple[np.ndarray | None, np.ndarray | None]:
    if matrix.shape[0] != matrix.shape[1] or matrix.shape[0] != vector.shape[0]:
        raise ShapeMismatchError
    if np.linalg.det(matrix) == 0:
        return None, None
    coordinates_in_the_basis = np.linalg.solve(matrix.T, vector.T)
    zero_matrix = np.zeros(shape = matrix.shape)
    #coordinates_of_projections = np.diag(coordinates_in_the_basis)
    coordinates_of_projections = coordinates_in_the_basis * matrix
    orthogonal_projections = vector[np.newaxis, :] - coordinates_of_projections

    print("coordinates_of_projections")
    print(coordinates_of_projections)
    print()

    print("basis")
    print(matrix)
    print()

    print("vector")
    print(vector)
    print()

    print("coordinates_in_the_basis")
    print(coordinates_in_the_basis)
    print()

    print("zero_matrix")
    print(zero_matrix)
    print()

    print("vector[np.newaxis, :]")
    print(vector[np.newaxis, :])
    print()

    print("orthogonal_projections")
    print(orthogonal_projections)
    print()
    return coordinates_of_projections, orthogonal_projections

matrix = np.diag([2, 3])
vector = np.arange(start=1, stop=3)
projections_expected = np.array([[1, 0], [0, 2]])
orthogonals_expected = np.array([[0, 2], [1, 0]])

projections, orthogonals = get_projections_components(matrix, vector)

print("res")
print()

print("res_projections")
print(projections)
print()

print("res_orthogonals")
print(orthogonals)
print()

print("projections_expected")
print(projections_expected)
print()

print("orthogonals_expected")
print(orthogonals_expected)
print()

assert np.allclose(projections, projections_expected)
assert np.allclose(orthogonals, orthogonals_expected)