Vectors, matrices and tensors: shapes and basic operations

The problem, from first principles

A single temperature reading is one number. A week of daily readings is a list of seven numbers. A month of readings across ten cities is a grid, ten rows, thirty-ish columns. A year of that, split by morning/afternoon/evening, stacks another axis on top of the grid. Nothing about the idea changes as you add axes, only how many numbers you need to address one value: none, one, two, three.

That single idea, "a number, or numbers indexed by however many axes," is the entire foundation this curriculum builds on. Before touching gradients or neural networks, you need the vocabulary for describing the shape of data and the two operations simple enough to define with no ambiguity at all: adding two same-shaped things, and multiplying them position by position.

From theory to code

Theory names the ranks (scalar, vector, matrix, tensor) and the two attributes every one of them has: shape (how many entries along each axis) and ndim (how many axes there are). Implement four small functions that expose this vocabulary directly, plus the two operations that only ever touch matching positions.

Implement shape_of, ndim_of, elementwise_add and elementwise_multiply against that reasoning. The signatures and docstrings are already in the editor.

Constraints

• shape_of returns a plain tuple, e.g. (3,) for a length-3 vector, () for a scalar.
• ndim_of returns a plain int.
• elementwise_add and elementwise_multiply assume a and b already share a shape (broadcasting is a later question's job).
• elementwise_multiply is NOT matrix multiplication: it must never reduce a vector down to a scalar.
• No Python loop over positions anywhere, this is meant to be vectorized.

Hints

Open one at a time. Each gives away a little more than the last.