Weighted Sum

Definition

A weighted sum multiplies each feature by a weight and adds the results.

Example:

features = [x1, x2]
weights  = [w1, w2]

Weighted sum:

w1*x1 + w2*x2

Weighted sum = add feature values after weighting them.

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Simple Example

Let:

features = [3, 4]
weights  = [2, 5]

Compute:

2*3 + 5*4
= 6 + 20
= 26

The weighted sum is:

26

Weighted sums turn a feature vector into one score.

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Dot Product Form

A weighted sum is a dot product.

Example:

w = [2, 5]
x = [3, 4]

Then:

dot(w, x) = 2*3 + 5*4 = 26

So:

weighted sum = dot(weights, features)

Weighted sums connect linear classification to vector geometry.

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Adding Bias

A linear score often adds bias after the weighted sum.

Example:

score = dot(w, x) + b

If:

dot(w, x) = 26
b = -10

then:

score = 16

Bias shifts the weighted sum into a decision score.

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Contribution Of Each Feature

Each term contributes to the total score.

Example:

2*3 + 5*4

Terms:

2*3 = 6
5*4 = 20

The second feature contributes more in this example.

A weighted sum decomposes the score into feature contributions.

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Sign Matters

Positive and negative terms can push the score in different directions.

Example:

weights  = [2, -5]
features = [3, 4]

Weighted sum:

2*3 + (-5)*4
= 6 - 20
= -14

The second feature pushes the score downward.

Weight signs determine whether features support or oppose the score.

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Why Weighted Sum Matters

The perceptron starts with a weighted sum.

Basic shape:

input features -> weighted sum -> threshold -> class

So before understanding perceptrons, the weighted sum must be clear.

A perceptron decision begins with a weighted sum.

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Final Summary

• A weighted sum multiplies features by weights and adds.
• It produces one scalar score.
• It is equivalent to a dot product.
• Bias can be added after it.
• Each feature contributes one term.
• Weight signs matter.
• Perceptrons begin with weighted sums.