Why must data be vectorized?

A neural net only accepts fixed-length numbers, not variable-length raw text. In the course spam inbox, each email becomes three counts: promotional words, links, unknown sender. Email A is $(2, 0, 1)$, email B is $(0, 1, 1)$ — same dimension $d = 3$.

Storing only the string "50% off today" leaves every message a different length, so there is no shared matrix multiply. Vectorization picks a feature dictionary and counts in a fixed order. Permuting columns changes meaning: $(2,0,1)$ is not $(0,2,1)$.

Python on four tiny emails:

counts = [(2, 0, 1), (0, 1, 1), (5, 2, 1), (0, 0, 0)]
assert all(len(x) == 3 for x in counts)

During batch training, NumPy stacks an $X$ matrix of shape (batch, 3) and multiplies once by $\mathbf{W}$. That is why deep learning courses start with vectorization: it is the prerequisite for GPUs and gradient descent at scale. The spam filter project in the course never sees raw email bodies in the training loop — only these fixed-length vectors.