Place holder 1

dot product

import numpy as np def matrix_dot_vector(a: list[list[int|float]], b: list[int|float]) -> list[int|float]: # Return a list where each element is the dot product of a row of ‘a’ with ‘b’. # If the number of columns in ‘a’ does not match the length of ‘b’, return -1. a = np.array(a) b = np.array(b) if a.shape[1]==b.shape[0]: return np.dot(a,b) else: return -1

pass

key learnings: np.shape, np.dot add content about the theory i wrote,

transpose: import numpy as np def transpose_matrix(a: list[list[int|float]]) -> list[list[int|float]]: b = np.transpose(np.array(a)) return b

key learnings: np.transpose

scalar multiplication: import numpy as np def scalar_multiply(matrix: list[list[int|float]], scalar: int|float) -> list[list[int|float]]: result = scalar * np.array(matrix) return result

Cosine similarity: import numpy as np

def cosine_similarity(v1, v2): # Implement your code here

v1 = np.array(v1)
v2 = np.array(v2)

result = (np.dot(v1,v2))/((np.linalg.norm(v1))*(np.linalg.norm(v2)))
return result
pass

key learnings: np.linalg.norm, np.dot The magnitude of a vector is also known as L2 norm:

The magnitude of a vector is known as the L2 norm because it's derived from the more general formula for p-norms, and the L2 norm is calculated using the value '\(p=2\)'. This specific value results in the familiar calculation of the square root of the sum of the squares of the vector's components, which corresponds to the Euclidean distance—the standard "straight-line" distance we use in everyday life. The p-norm family: A "norm" is a function that measures the length or size of a vector. The general formula for a p-norm is the \(p\)-th root of the sum of the absolute values of each component raised to the power of \(p\).Substituting p = 2: When you substitute \(p=2\) into this general formula, you get the L2 norm:The L2 norm of a vector \(\vec{x}\) is calculated as \((\sum _{i=1}^{n}|x_{i}|^{2})^{1/2}\)