Notation to Array Idioms

The dot product of two equal-length vectors multiplies matching entries and adds them up: $a\cdot b=\sum_i a_ib_i$ . It is the single most common operation in a neural network — every neuron's pre-activation is one.

`a`	`b`	result
`[1, 2, 3]`	`[4, 5, 6]`	`32`
`[2, -1]`	`[3, 4]`	`2`
`[1, 0]`	`[0, 1]`	`0`

One summed index and nothing left over: $\sum_i a_ib_i$ runs $i$ over every position, so the result is a scalar, not a vector. A result of $0$ means the vectors are perpendicular.

No loop needed: in NumPy a @ b (1-D times 1-D is the dot product), in Eigen a.dot(b).

Python

def dot(a: np.ndarray, b: np.ndarray) -> float:
    """Dot product of two equal-length vectors: sum_i a_i * b_i.

    a: [D]
    b: [D]
    Returns a scalar float
    """
    return float(a @ b)

Dot product

Matrix × vector