# Path: scratch/linear_algebra.py
# def add(v: Vector, w: Vector) -> Vector:
# def subtract(v: Vector, w: Vector) -> Vector:
# def vector_sum(vectors: List[Vector]) -> Vector:
# def scalar_multiply(c: float, v: Vector) -> Vector:
# def vector_mean(vectors: List[Vector]) -> Vector:
# def dot(v: Vector, w: Vector) -> float:
# def sum_of_squares(v: Vector) -> float:
# def magnitude(v: Vector) -> float:
# def squared_distance(v: Vector, w: Vector) -> float:
# def distance(v: Vector, w: Vector) -> float:
# def distance(v: Vector, w: Vector) -> float:  # type: ignore
# def shape(A: Matrix) -> Tuple[int, int]:
# def get_row(A: Matrix, i: int) -> Vector:
# def get_column(A: Matrix, j: int) -> Vector:
# def make_matrix(num_rows: int,
#                 num_cols: int,
#                 entry_fn: Callable[[int, int], float]) -> Matrix:
# def identity_matrix(n: int) -> Matrix:
# A = [[1, 2, 3],  # A has 2 rows and 3 columns
#      [4, 5, 6]]
# B = [[1, 2],     # B has 3 rows and 2 columns
#      [3, 4],
#      [5, 6]]
# 
# Path: scratch/linear_algebra.py
# def distance(v: Vector, w: Vector) -> float:
#     """Computes the distance between v and w"""
#     return math.sqrt(squared_distance(v, w))
# 
# def add(v: Vector, w: Vector) -> Vector:
#     """Adds corresponding elements"""
#     assert len(v) == len(w), "vectors must be the same length"
# 
#     return [v_i + w_i for v_i, w_i in zip(v, w)]
# 
# def scalar_multiply(c: float, v: Vector) -> Vector:
#     """Multiplies every element by c"""
#     return [c * v_i for v_i in v]


from scratch.linear_algebra import Vector, dot
from typing import Callable
from scratch.linear_algebra import distance, add, scalar_multiply
from typing import TypeVar, List, Iterator
    from scratch.linear_algebra import vector_mean
import random
    import matplotlib.pyplot as plt

def sum_of_squares(v: Vector) -> float:
    """Computes the sum of squared elements in v"""
    return dot(v, v)


def difference_quotient(f: Callable[[float], float],
                        x: float,
                        h: float) -> float:
    return (f(x + h) - f(x)) / h

def square(x: float) -> float:
    return x * x

def derivative(x: float) -> float:
    return 2 * x

def estimate_gradient(f: Callable[[Vector], float],
                      v: Vector,
                      h: float = 0.0001):
    return [partial_difference_quotient(f, v, i, h)
            for i in range(len(v))]


def gradient_step(v: Vector, gradient: Vector, step_size: float) -> Vector:
    """Moves `step_size` in the `gradient` direction from `v`"""
    assert len(v) == len(gradient)
    step = scalar_multiply(step_size, gradient)

# Path: scratch/linear_algebra.py
# def add(v: Vector, w: Vector) -> Vector:
# def subtract(v: Vector, w: Vector) -> Vector:
# def vector_sum(vectors: List[Vector]) -> Vector:
# def scalar_multiply(c: float, v: Vector) -> Vector:
# def vector_mean(vectors: List[Vector]) -> Vector:
# def dot(v: Vector, w: Vector) -> float:
# def sum_of_squares(v: Vector) -> float:
# def magnitude(v: Vector) -> float:
# def squared_distance(v: Vector, w: Vector) -> float:
# def distance(v: Vector, w: Vector) -> float:
# def distance(v: Vector, w: Vector) -> float:  # type: ignore
# def shape(A: Matrix) -> Tuple[int, int]:
# def get_row(A: Matrix, i: int) -> Vector:
# def get_column(A: Matrix, j: int) -> Vector:
# def make_matrix(num_rows: int,
#                 num_cols: int,
#                 entry_fn: Callable[[int, int], float]) -> Matrix:
# def identity_matrix(n: int) -> Matrix:
# A = [[1, 2, 3],  # A has 2 rows and 3 columns
#      [4, 5, 6]]
# B = [[1, 2],     # B has 3 rows and 2 columns
#      [3, 4],
#      [5, 6]]
# 
# Path: scratch/linear_algebra.py
# def distance(v: Vector, w: Vector) -> float:
#     """Computes the distance between v and w"""
#     return math.sqrt(squared_distance(v, w))
# 
# def add(v: Vector, w: Vector) -> Vector:
#     """Adds corresponding elements"""
#     assert len(v) == len(w), "vectors must be the same length"
# 
#     return [v_i + w_i for v_i, w_i in zip(v, w)]
# 
# def scalar_multiply(c: float, v: Vector) -> Vector:
#     """Multiplies every element by c"""
#     return [c * v_i for v_i in v]


# Path: scratch/gradient_descent.py
from scratch.linear_algebra import Vector, dot
from typing import Callable
from scratch.linear_algebra import distance, add, scalar_multiply
from typing import TypeVar, List, Iterator
    from scratch.linear_algebra import vector_mean
import random
    import matplotlib.pyplot as plt


def sum_of_squares(v: Vector) -> float:
    """Computes the sum of squared elements in v"""
    return dot(v, v)


def difference_quotient(f: Callable[[float], float],
                        x: float,
                        h: float) -> float:
    return (f(x + h) - f(x)) / h

def square(x: float) -> float:
    return x * x

def derivative(x: float) -> float:
    return 2 * x

def estimate_gradient(f: Callable[[Vector], float],
                      v: Vector,
                      h: float = 0.0001):
    return [partial_difference_quotient(f, v, i, h)
            for i in range(len(v))]


def gradient_step(v: Vector, gradient: Vector, step_size: float) -> Vector:
    """Moves `step_size` in the `gradient` direction from `v`"""
    assert len(v) == len(gradient)
    step = scalar_multiply(step_size, gradient)

    return add(v, step)