Math Lab · ML to LLMs

Math Lab

Change the inputs and every intermediate step recomputes. Use it to check your hand calculations from the worked examples.

Linear algebra

Dot product

a \cdot b = \sum_{i=1}^{n} a_i b_i

Symbols
$a, b$	vectors of the same length n
$a_i$	the i-th entry of a
$n$	number of entries

InputsaModule 0 example: 2 bedrooms, 1.5 thousand sq ftbweights: $k per bedroom, $k per thousand sq ft

1
Multiply matching entries
$2 \times 30 + 1.5 \times 120$
2
Evaluate each product
$60 + 180$
3
Add them up
$= 240$

Result

a \cdot b = 240

A weighted sum: each entry of a is scaled by the matching weight in b. Geometrically it equals ‖a‖‖b‖cos θ = 2.5 × 123.6932 × cos θ, so its sign tells you whether the vectors point the same way.

Common mistakes

Mixing up the dot product (one number) with element-wise multiplication (a vector).
Forgetting that lengths must match. In code, a shape error here usually means a transposed matrix upstream.

Related lesson: Vectors and matrices