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linear_algebra.py
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linear_algebra.py
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from typing import List
Vector = List[float]
height_weight_age = [70, # inches,
170, # pounds,
40 ] # years
grades = [95, # exam1
80, # exam2
75, # exam3
62 ] # exam4
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)]
assert add([1, 2, 3], [4, 5, 6]) == [5, 7, 9]
def subtract(v: Vector, w: Vector) -> Vector:
"""Subtracts 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)]
assert subtract([5, 7, 9], [4, 5, 6]) == [1, 2, 3]
def vector_sum(vectors: List[Vector]) -> Vector:
"""Sums all corresponding elements"""
# Check that vectors is not empty
assert vectors, "no vectors provided!"
# Check the vectors are all the same size
num_elements = len(vectors[0])
assert all(len(v) == num_elements for v in vectors), "different sizes!"
# the i-th element of the result is the sum of every vector[i]
return [sum(vector[i] for vector in vectors)
for i in range(num_elements)]
assert vector_sum([[1, 2], [3, 4], [5, 6], [7, 8]]) == [16, 20]
def scalar_multiply(c: float, v: Vector) -> Vector:
"""Multiplies every element by c"""
return [c * v_i for v_i in v]
assert scalar_multiply(2, [1, 2, 3]) == [2, 4, 6]
def vector_mean(vectors: List[Vector]) -> Vector:
"""Computes the element-wise average"""
n = len(vectors)
return scalar_multiply(1/n, vector_sum(vectors))
assert vector_mean([[1, 2], [3, 4], [5, 6]]) == [3, 4]
def dot(v: Vector, w: Vector) -> float:
"""Computes v_1 * w_1 + ... + v_n * w_n"""
assert len(v) == len(w), "vectors must be same length"
return sum(v_i * w_i for v_i, w_i in zip(v, w))
assert dot([1, 2, 3], [4, 5, 6]) == 32 # 1 * 4 + 2 * 5 + 3 * 6
def sum_of_squares(v: Vector) -> float:
"""Returns v_1 * v_1 + ... + v_n * v_n"""
return dot(v, v)
assert sum_of_squares([1, 2, 3]) == 14 # 1 * 1 + 2 * 2 + 3 * 3
import math
def magnitude(v: Vector) -> float:
"""Returns the magnitude (or length) of v"""
return math.sqrt(sum_of_squares(v)) # math.sqrt is square root function
assert magnitude([3, 4]) == 5
def squared_distance(v: Vector, w: Vector) -> float:
"""Computes (v_1 - w_1) ** 2 + ... + (v_n - w_n) ** 2"""
return sum_of_squares(subtract(v, w))
def distance(v: Vector, w: Vector) -> float:
"""Computes the distance between v and w"""
return math.sqrt(squared_distance(v, w))
def distance(v: Vector, w: Vector) -> float: # type: ignore
return magnitude(subtract(v, w))
# Another type alias
Matrix = List[List[float]]
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]]
from typing import Tuple
def shape(A: Matrix) -> Tuple[int, int]:
"""Returns (# of rows of A, # of columns of A)"""
num_rows = len(A)
num_cols = len(A[0]) if A else 0 # number of elements in first row
return num_rows, num_cols
assert shape([[1, 2, 3], [4, 5, 6]]) == (2, 3) # 2 rows, 3 columns
def get_row(A: Matrix, i: int) -> Vector:
"""Returns the i-th row of A (as a Vector)"""
return A[i] # A[i] is already the ith row
def get_column(A: Matrix, j: int) -> Vector:
"""Returns the j-th column of A (as a Vector)"""
return [A_i[j] # jth element of row A_i
for A_i in A] # for each row A_i
from typing import Callable
def make_matrix(num_rows: int,
num_cols: int,
entry_fn: Callable[[int, int], float]) -> Matrix:
"""
Returns a num_rows x num_cols matrix
whose (i,j)-th entry is entry_fn(i, j)
"""
return [[entry_fn(i, j) # given i, create a list
for j in range(num_cols)] # [entry_fn(i, 0), ... ]
for i in range(num_rows)] # create one list for each i
def identity_matrix(n: int) -> Matrix:
"""Returns the n x n identity matrix"""
return make_matrix(n, n, lambda i, j: 1 if i == j else 0)
assert identity_matrix(5) == [[1, 0, 0, 0, 0],
[0, 1, 0, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 0],
[0, 0, 0, 0, 1]]
data = [[70, 170, 40],
[65, 120, 26],
[77, 250, 19],
# ....
]
friendships = [(0, 1), (0, 2), (1, 2), (1, 3), (2, 3), (3, 4),
(4, 5), (5, 6), (5, 7), (6, 8), (7, 8), (8, 9)]
# user 0 1 2 3 4 5 6 7 8 9
#
friend_matrix = [[0, 1, 1, 0, 0, 0, 0, 0, 0, 0], # user 0
[1, 0, 1, 1, 0, 0, 0, 0, 0, 0], # user 1
[1, 1, 0, 1, 0, 0, 0, 0, 0, 0], # user 2
[0, 1, 1, 0, 1, 0, 0, 0, 0, 0], # user 3
[0, 0, 0, 1, 0, 1, 0, 0, 0, 0], # user 4
[0, 0, 0, 0, 1, 0, 1, 1, 0, 0], # user 5
[0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 6
[0, 0, 0, 0, 0, 1, 0, 0, 1, 0], # user 7
[0, 0, 0, 0, 0, 0, 1, 1, 0, 1], # user 8
[0, 0, 0, 0, 0, 0, 0, 0, 1, 0]] # user 9
assert friend_matrix[0][2] == 1, "0 and 2 are friends"
assert friend_matrix[0][8] == 0, "0 and 8 are not friends"
# only need to look at one row
friends_of_five = [i
for i, is_friend in enumerate(friend_matrix[5])
if is_friend]