Weighted Sum

Definition

A weighted sum combines inputs by multiplying each input by a weight, then adding the results.

Basic form:

weighted sum = w1*x1 + w2*x2 + w3*x3

Here:

x values = inputs
w values = weights

A weighted sum is a controlled combination of inputs.

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Why Weights Matter

Weights control how strongly each input affects the result.

Example:

x1 = 10
x2 = 10

w1 = 2
w2 = 0.5

Then:

w1*x1 = 20
w2*x2 = 5

Even though the inputs are equal, the first input affects the score more.

A weight assigns influence to an input.

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Positive, Negative, and Zero Weights

Weights can affect the sum in different ways.

positive weight -> input pushes score upward
negative weight -> input pushes score downward
zero weight -> input has no effect

Example:

score = 2*x1 - 3*x2

x1 increases the score.

x2 decreases the score.

The sign and size of a weight control direction and strength.

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Weighted Sum in a Neuron

A neuron usually begins by computing a weighted sum.

inputs -> weighted sum -> activation -> output

The weighted sum creates a score from the inputs.

The activation function then transforms that score.

The weighted sum is the neuron's first combination step.

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Learning Changes Weights

Weights are parameters.

During training, optimization changes them to reduce loss.

Example:

old weights -> output -> high loss
updated weights -> output -> lower loss

The network learns by changing how strongly inputs are combined.

Learning changes the influence of inputs by changing weights.

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Final Summary

• A weighted sum multiplies inputs by weights and adds the results.
• Weights control input influence.
• Positive weights increase the score.
• Negative weights decrease the score.
• Zero weights remove input influence.
• Neural networks learn by adjusting weights.