Per-Row with reshape(-1, 1)

Now flip the problem: each player has a form rating (5 values), and you want to scale every match in that player's row by his own rating. Virat 1.15, Rohit 1.0, Hardik 0.95, Shubman 1.05, Shreyas 1.1. This is the broadcasting case that confuses almost every beginner — Munna calls it "the tricky one".

Definition

A column vector is a 2D array of shape (n, 1) — n rows, one column. Combining a column vector with a 2D array of shape (n, cols) stretches the single column across every column, so each value is applied to one whole row.

• (n, 1) vs (n,) — they hold the same n numbers, but (n,) is 1D and aligns with the last dimension, while (n, 1) is 2D and aligns its n with the rows.
• reshape(-1, 1) — "make it one column, and work out (-1) the number of rows yourself".
• One value per row — in the squad table, rows are players, so this gives a per-player adjustment.

Why the plain 1D array fails

import numpy as np

squad_scores = np.array([
    [72, 45, 88, 31, 64,  0],   # Virat
    [56,  0, 74, 88, 32, 45],   # Rohit
    [45, 32,  0, 67, 18, 54],   # Hardik
    [33, 61, 45,  0, 77, 83],   # Shubman
    [88, 14, 57, 72,  0, 38],   # Shreyas
])
player_form_rating = np.array([1.15, 1.0, 0.95, 1.05, 1.1])
squad_scores * player_form_rating

ValueError: operands could not be broadcast together with shapes (5,6) (5,)

squad_scores        5  6
player_form_rating     5     -> (1, 5)
compare right-to-left: 6 vs 5  -> FAIL

The 5 matches the rows, but NumPy never looks there first — it always aligns from the right.

Syntax — three ways to make a column vector

col = arr.reshape(-1, 1)            # most common
col = arr[:, np.newaxis]            # insert a new axis of size 1 at position 1
col = arr[:, None]                  # same thing (np.newaxis is None)
col = np.expand_dims(arr, axis=1)   # function form

Example — the fix: per-player form rating

col = player_form_rating.reshape(-1, 1)
print(col.shape)
print(col)
print((squad_scores * col).round(1))

(5, 1)
[[1.15]
 [1.  ]
 [0.95]
 [1.05]
 [1.1 ]]
[[ 82.8  51.7 101.2  35.6  73.6   0. ]
 [ 56.    0.   74.   88.   32.   45. ]
 [ 42.8  30.4   0.   63.6  17.1  51.3]
 [ 34.6  64.   47.2   0.   80.9  87.2]
 [ 96.8  15.4  62.7  79.2   0.   41.8]]

How it works:

squad_scores   5  6
col            5  1
               6 vs 1 -> stretch the column across 6 matches
               5 vs 5 -> equal
Result: (5, 6)

Virat's whole row is multiplied by 1.15 (72 → 82.8), Rohit's row by 1.0 (unchanged), Hardik's by 0.95 and so on.

Example — compare each score with the player's own average

Aggregating with axis=1 gives one value per row, shape (5,) — which does not broadcast back. Reshape it, or use keepdims=True:

player_avg = squad_scores.mean(axis=1)
print(player_avg.round(2))
centered = squad_scores - player_avg.reshape(-1, 1)
print(centered.round(1))
print(squad_scores.mean(axis=1, keepdims=True).shape)

[50.   49.17 36.   49.83 44.83]
[[ 22.   -5.   38.  -19.   14.  -50. ]
 [  6.8 -49.2  24.8  38.8 -17.2  -4.2]
 [  9.   -4.  -36.   31.  -18.   18. ]
 [-16.8  11.2  -4.8 -49.8  27.2  33.2]
 [ 43.2 -30.8  12.2  27.2 -44.8  -6.8]]
(5, 1)

How it works: keepdims=True keeps the reduced axis with size 1, so the result is already (5, 1) — ready to broadcast. squad_scores - squad_scores.mean(axis=1, keepdims=True) is the cleanest one-liner.

Example — column + row = a full table

Combining a column vector with a row (1D) array makes both stretch — a quick way to build a table of every combination. Runs scored after 1–5 overs at run rates 6, 8 and 10:

overs = np.arange(1, 6)          # (5,)
rr = np.array([6, 8, 10])        # (3,)
print(overs.reshape(-1, 1) * rr)

[[ 6  8 10]
 [12 16 20]
 [18 24 30]
 [24 32 40]
 [30 40 50]]

(5, 1) × (3,) → (5, 3): row = overs, column = run rate.

Mental shortcut

Common mistakes

• ❌ scores * ratings with ratings.shape == (rows,). ✅ scores * ratings.reshape(-1, 1).
• ❌ reshape(1, -1) when you wanted a column. ✅ (1, -1) is a row; (-1, 1) is a column.
• ❌ scores - scores.mean(axis=1). ✅ scores - scores.mean(axis=1, keepdims=True).

Interview tip

Q: What is the difference between shapes (5,), (5, 1) and (1, 5)? A: (5,) is a 1D array — it aligns with the last dimension. (5, 1) is a column vector (5 rows, 1 column) — it broadcasts across columns, one value per row. (1, 5) is a row vector — it broadcasts down rows, one value per column, behaving like (5,).