Vectors & Matrices

A vector is just an ordered list of numbers — [3, 4] can represent a point, a direction, a set of feature weights, anything with multiple components. Two operations you'll use constantly:

Vector addition (component-wise): [1, 2] + [3, 4] = [4, 6]

Scalar multiplication: 2 * [1, 2] = [2, 4]

The dot product multiplies corresponding components and sums them — it's a single number (a "scalar") that captures how much two vectors point in the same direction:

def dot_product(a, b):
    return sum(x * y for x, y in zip(a, b))

dot_product([1, 2, 3], [4, 5, 6])   # 1*4 + 2*5 + 3*6 = 32

You already met this exact computation in the NumPy course (np.dot) — this course is about understanding what it means and why it works; NumPy is about computing it efficiently at scale. Both matter.

A matrix is a grid of numbers — a list of rows, each row a list of numbers of the same length. [[1, 2], [3, 4]] is a 2×2 matrix. The transpose of a matrix flips it over its diagonal, turning rows into columns:

def transpose(m):
    return [[row[i] for row in m] for i in range(len(m[0]))]

transpose([[1, 2, 3], [4, 5, 6]])
# [[1, 4], [2, 5], [3, 6]]

The identity matrix (1s on the diagonal, 0s elsewhere) is the matrix equivalent of the number 1 — multiplying any matrix by it leaves the matrix unchanged, the same way multiplying any number by 1 does.