Linear fwd

The problem, from first principles

Every neural network, no matter how large or how exotic its attention mechanisms or convolutions, is built out of one operation repeated over and over: a linear (fully connected / dense) transformation of its input, usually followed by a nonlinearity. Understanding what a "linear layer" actually computes, down to the exact shapes involved, is the foundation everything else in this Part builds on: [02-layers/02-linear-backward] needs this forward pass to differentiate, and the Sequential container you'll build later in this track needs to chain many of these together correctly.

A linear layer holds two learnable tensors: a weight matrix and a bias vector. Given an input vector, it computes a new vector where each output entry is a weighted sum of every input entry, plus a learned constant offset. Stacked into a batch of many input vectors at once (the normal case during training, for GPU efficiency), this becomes a single matrix multiplication.

From theory to code

Implement linear_forward(x, weight, bias). x has shape (batch_size, in_features). Match PyTorch's own nn.Linear convention exactly: weight has shape (out_features, in_features) (each ROW of weight is the set of weights for one output neuron), not (in_features, out_features). bias has shape (out_features,). The output has shape (batch_size, out_features).

Constraints

• weight is (out_features, in_features), matching torch.nn.Linear.weight's shape exactly, not its transpose.
• The bias must broadcast correctly across the batch dimension: every row of the output gets the same bias vector added.
• Must work for any batch_size, including batch_size == 1.

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