Weighted Scoring with @

Matrix multiplication is the backbone of linear regression, neural networks and recommendation systems — but you can understand why with a simple fantasy-cricket problem. A fantasy platform gives 1 point per run, 25 points per wicket and 10 points per catch. We want every player's total, in one operation.

Definition

A weighted sum multiplies each value by a weight (how much it counts) and adds the results: score = runs×1 + wickets×25 + catches×10. Computing a weighted sum for every row of a data matrix at once is exactly a matrix–vector product: stats @ weights.

• Stats matrix — shape (players, features): one row per player, one column per stat.
• Weights — one number per feature (column), shape (features,) or (features, 1).
• Result — one score per player, shape (players,) or (players, 1).

This is literally how a linear model predicts: predictions = X @ w, where X holds the features and w the learned weights.

Syntax

scores = stats @ weights            # weights 1D (k,)  -> scores (n,)
scores = stats @ weights_col        # weights (k, 1)   -> scores (n, 1)
scores = (stats * weights).sum(axis=1)   # same numbers, via broadcasting

Example — fantasy points for one match

import numpy as np
PLAYERS = ["Virat", "Rohit", "Hardik", "Shubman", "Shreyas"]
match_stats = np.array([
    [72, 0, 1],   # Virat:   72 runs, 0 wickets, 1 catch
    [56, 0, 0],   # Rohit:   56 runs, 0 wickets, 0 catches
    [12, 2, 1],   # Hardik:  12 runs, 2 wickets, 1 catch
    [33, 0, 2],   # Shubman: 33 runs, 0 wickets, 2 catches
    [45, 1, 0],   # Shreyas: 45 runs, 1 wicket,  0 catches
])
point_values = np.array([[1], [25], [10]])   # per run, per wicket, per catch

fantasy_points = match_stats @ point_values
print("shapes:", match_stats.shape, "@", point_values.shape, "->", fantasy_points.shape)
for name, pts in zip(PLAYERS, fantasy_points[:, 0]):
    print(f"  {name:<8} {pts:>4} points")
print("Hardik by hand:", 12*1 + 2*25 + 1*10)

shapes: (5, 3) @ (3, 1) -> (5, 1)
  Virat      82 points
  Rohit      56 points
  Hardik     72 points
  Shubman    53 points
  Shreyas    70 points
Hardik by hand: 72

How it works: (5, 3) @ (3, 1) → (5, 1). Each player's row [runs, wickets, catches] is dotted with the column [1, 25, 10]. Hardik: 12×1 + 2×25 + 1×10 = 72. One line computes all five players — or five million.

Example — 1D weights, and the broadcasting equivalent

weights = np.array([1, 25, 10])        # 1D vector
print(match_stats @ weights)           # shape (5,)
print((match_stats * weights).sum(axis=1))

[82 56 72 53 70]
[82 56 72 53 70]

With a 1D weight vector you get a flat 1D result, which is usually more convenient. The second line does the same maths in two steps (broadcast multiply, then sum each row) — useful to understand, but @ is shorter and faster.

Example — compare two scoring systems at once

schemes = np.array([[1, 1],      # points per run   (scheme A, scheme B)
                    [25, 30],    # points per wicket
                    [10, 8]])    # points per catch
print(match_stats @ schemes)

[[82 80]
 [56 56]
 [72 80]
 [53 49]
 [70 75]]

(5, 3) @ (3, 2) → (5, 2): column 0 is scheme A, column 1 is scheme B. Scheme B (more for wickets, less for catches) helps the all-rounders Hardik and Shreyas. Each extra column of weights = one extra column of results, still a single operation.

Where this pattern appears

Common mistakes

• ❌ Weights in the wrong order ([25, 1, 10] when columns are runs, wickets, catches). ✅ The i-th weight must match the i-th column.
• ❌ match_stats * point_values with point_values of shape (3, 1) → broadcasting error or wrong shape. ✅ Use @ for weighted sums.
• ❌ Forgetting that a (k, 1) weight column gives a (n, 1) result — then print(result[2]) shows [72], not 72. ✅ Use a 1D weight vector, or .ravel() the result.

Interview tip

Q: How is a linear-regression prediction computed in NumPy? A: y_pred = X @ w + b — the feature matrix X of shape (n_samples, n_features) times the weight vector w of shape (n_features,) gives one prediction per sample; the bias b is added by broadcasting.