One example, one dot product
For features x = [2, 3] and weights w = [0.5, −1], the score is w · x + b = 0.5 × 2 − 1 × 3 + b. The dot product combines corresponding entries; the order and units of features must match the weights.
A layer as a matrix
If W has shape (outputs, inputs) and x has shape (inputs,), then Wx + b gives one value per output. Each row of W is a different set of weights. A batch can be represented differently depending on convention, so check the shapes rather than memorizing a single layout. Try changing just one input at a time. Each bar is that input multiplied by its fixed weight, and adding the contributions produces the dot product.
INTERACTIVE EXPLORATION
Watch a dot product change
Change one component of the input vector. The two bars are the contributions from each feature to w · x.
A geometric view
In two dimensions, a vector can be drawn as an arrow. Linear transformations move and reshape space while preserving addition and scaling. The same algebra applies in higher dimensions even when the geometry is harder to draw.
Check your understanding
If W has 3 rows and 2 columns and x has 2 entries, how many outputs does Wx have?
Three: one output for each row of W.