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Tensor math splits into two families: elementwise ops that keep the shape, and reductions that collapse it down to fewer numbers.

An elementwise op applies the same action to every entry independently. Input shape in, same shape out, no mixing between positions.

With two tensors of equal shape, elementwise ops act position by position. Add matches index to index, multiply does the same.

Functions like torch.relu, exp, and sqrt run elementwise. Each number is transformed on its own, and the tensor keeps its original shape.

A reduction combines many values into fewer. Sum, mean, and max fold a whole tensor down, by default to a single scalar.

Pass dim to reduce along one axis only. The chosen axis disappears while the others stay, so a 2D tensor becomes 1D.

Set keepdim=True to keep the reduced axis as size 1. That preserved shape is what makes later broadcasting line up cleanly.

mean divides, so it expects floating-point input. Call it on an integer tensor and PyTorch will complain until you cast to float first.

Sometimes you want the position, not the value. argmax returns the index of the largest entry, which is how a classifier picks its predicted class.

Real code stacks both kinds: an elementwise transform feeds a reduction. Square the errors, then take the mean, and you have mean squared error.

The rule of thumb: reach for elementwise when every value should change in place, and reach for a reduction when you need a summary like a total or average.

Can you tell which operation changes a tensor shape?

Elementwise ops keep the shape and act per value; reductions like sum and mean collapse it, with dim and keepdim controlling exactly how. 🎯