Matrix Broadcasting

When performing operations on arrays with different shapes, numpy automatically expands one array to match the other. This is broadcasting.

Rules:

1. Compare dimensions from right to left
2. Dimensions must be equal, or one must be 1
3. If either is 1, expand to match the other

Example:

arr = np.array([1, 2, 3, 4])
arr * 2  # [2, 4, 6, 8]
# Expands [2] to [2, 2, 2, 2]

Error case:

a = np.array([1, 2, 3])
b = np.array([1, 2])
a + b  # ValueError: shapes (3,) and (2,) not aligned