Vector Operations

Implement fundamental vector and matrix operations from scratch. These operations are the building blocks of machine learning.

Functions to implement

1. dot_product(a, b)

Compute the dot product of two vectors.

• Input: Two lists of numbers of equal length
• Output: A single number (the sum of element-wise products)

2. vector_add(a, b)

Add two vectors element-wise.

• Input: Two lists of numbers of equal length
• Output: A list with element-wise sums

3. scalar_multiply(scalar, v)

Multiply a vector by a scalar.

• Input: A scalar and a list of numbers
• Output: A list with each element multiplied by the scalar

4. vector_norm(v)

Compute the L2 (Euclidean) norm of a vector.

• Input: A list of numbers
• Output: The square root of the sum of squared elements

5. matrix_vector_multiply(M, v)

Multiply a matrix by a vector.

• Input: A 2D list (matrix) and a 1D list (vector)
• Output: A list representing the result

Examples

dot_product([1, 2, 3], [4, 5, 6])  # 32
vector_add([1, 2], [3, 4])         # [4, 6]
scalar_multiply(2, [1, 2, 3])      # [2, 4, 6]
vector_norm([3, 4])                # 5.0
matrix_vector_multiply([[1, 2], [3, 4]], [5, 6])  # [17, 39]

Notes

• Do not use NumPy or other libraries
• Implement using only basic Python operations
• You may use math.sqrt for the square root