from typing import List, Tuple
import math

Matrix = List[List[float]]
Vector = Tuple[float, float]

def transpose(m: Matrix) -> Matrix:
    if not m or not m[0]:
        raise ValueError("Matrix must be non-empty")
    return [list(row) for row in zip(*m)]

def matmul(a: Matrix, b: Matrix) -> Matrix: #Matrixmultiplikation#
    if not a or not b or not a[0] or not b[0]:
        raise ValueError("Matrices must be non-empty")
    m, n = len(a), len(a[0])
    n2, p = len(b), len(b[0])
    if n != n2:
        raise ValueError("Incompatible shapes")
    c: Matrix = [[0.0 for _ in range(p)] for _ in range(m)]
    for i in range(m):
        for k in range(n):
            a_ik = float(a[i][k])
            for j in range(p):
                c[i][j] += a_ik * float(b[k][j])
def rotation_matrix(theta: float) -> Matrix:
    """2D-Rotationsmatrix R(theta)."""
    c, s = math.cos(theta), math.sin(theta)
    return [[c, -s],
            [s,  c]]

def apply_mat_to_vec(m: Matrix, v: Vector) -> Vector:
    """Wendet eine 2×2-Matrix auf einen 2D-Vektor (als Spaltenvektor) an."""
    if len(m) != 2 or len(m[0]) != 2:
        raise ValueError("Matrix must be 2x2 for 2D vector transform")
    x, y = float(v[0]), float(v[1])
    x2 = m[0][0]*x + m[0][1]*y
    y2 = m[1][0]*x + m[1][1]*y
    return (x2, y2)

def add(v1: Vector, v2: Vector) -> Vector:
    """Vektor-Addition v1 + v2."""
    return (v1[0] + v2[0], v1[1] + v2[1])