Problem: When Broadcasting Doesn't Broadcast

You have a 2D array and want to add a 1D array to each row. Simple, right? Just use broadcasting. Except sometimes NumPy throws a shape mismatch error when you least expect it, and the error message doesn't exactly explain why.

The problem

Let's say you're working with some data and want to add a vector to each row of a matrix. This should work:

import numpy as np

# This works fine

arr = np.array([[1, 2, 3], [4, 5, 6]]) # shape: (2, 3)
vec = np.array([10, 20, 30]) # shape: (3,)
result = arr + vec # ✅ Works! Broadcasting magic
print(result)

[[11 22 33]
[14 25 36]]

Great! Broadcasting works. But then you try something similar and it fails:

# But this fails - why?
arr2 = np.array([[1, 2], [3, 4]])       # shape: (2, 2)
vec2 = np.array([10, 20, 30])          # shape: (3,)
result2 = arr2 + vec2                   # ❌ ValueError: operands could not be broadcast

The error message says "operands could not be broadcast" but doesn't explain why. Both arrays have compatible-looking shapes, right? Wrong.

How broadcasting actually works

Broadcasting aligns dimensions from the right, not the left. When NumPy sees (2, 3) + (3,), it:

1. Looks at the rightmost dimensions: 3 matches 3 ✅
2. Treats the 1D array as if it has shape (1, 3)
3. Broadcasts along the first dimension: (2, 3) + (1, 3) → (2, 3)

But when you have (2, 2) + (3,):

1. Looks at the rightmost dimensions: 2 doesn't match 3 ❌
2. Stops. No broadcasting possible.

The fix (and when you need it)

If you need to add a vector to rows, make sure the trailing dimension matches. If it doesn't, reshape explicitly:

# Option 1: Reshape the vector to have a leading dimension
arr2 = np.array([[1, 2], [3, 4]])       # shape: (2, 2)
vec2 = np.array([10, 20])              # shape: (2,)
vec2_reshaped = vec2.reshape(1, -1)     # shape: (1, 2)
result = arr2 + vec2_reshaped          # ✅ Works! (2, 2) + (1, 2) → (2, 2)

# Option 2: Use np.newaxis (cleaner syntax)

result = arr2 + vec2[np.newaxis, :] # Same as reshape(1, -1)

# Option 3: Add a dimension with None (also same thing)

result = arr2 + vec2[None, :] # Most concise

The key insight

Broadcasting aligns from the right. Always check trailing dimensions first. If you're getting shape mismatch errors, print the shapes with .shape and check if the rightmost dimensions match. If they don't, you'll need to reshape explicitly.

The rule: two arrays can be broadcast together if, starting from the right, each dimension either matches or one of them is 1 (or missing, which NumPy treats as 1).