7. Broadcasting

Broadcasting lets NumPy operate on arrays of different shapes without copying data, by virtually stretching smaller arrays.

The rules (compare shapes from the right)#

1. If dimensions differ in count, left-pad the smaller shape with 1s.
2. Two dimensions are compatible if they are equal or one of them is 1.
3. A size-1 dimension is stretched to match the other.
4. If any dimension is incompatible → ValueError.

flowchart TD
    A["A · shape (3, 4)"] --> C{"align shapes from the right"}
    B["B · shape (4,) → padded to (1, 4)"] --> C
    C -->|"last dim: 4 == 4 ✓ · next dim: 3 vs 1 → stretch"| D["B virtually stretched to (3, 4)"]
    D --> E["element-wise result · shape (3, 4)"]

A      (3, 4)
B         (4,)   -> treated as (1, 4) -> stretched to (3, 4)  ✅

A      (3, 1)
B      (1, 4)    -> both stretch -> (3, 4)  ✅

A      (3, 4)
B         (3,)   -> (1, 3) vs (3, 4): 3 != 4  ❌ ValueError

Examples#

a = np.array([[1], [2], [3]])   # shape (3, 1)
b = np.array([10, 20, 30])      # shape (3,)
a + b
# [[11 21 31]
#  [12 22 32]
#  [13 23 33]]                   # outer sum, shape (3, 3)

m = np.arange(12).reshape(3, 4)
col_mean = m.mean(axis=0)       # shape (4,)
centered = m - col_mean         # subtract column means (broadcast)

💡 Tip: To force a column vector for broadcasting: col = a.reshape(-1, 1) or a[:, None].

⭐ Interview Question: Explain broadcasting. NumPy aligns shapes from the right, stretches size-1 dimensions, and applies the op element-wise without materializing the stretched copies — saving memory and time.

Real-world use case: Normalizing an image: img - img.mean(axis=(0,1)) broadcasts per-channel means across all pixels.