Backpropagation through a linear (fully-connected) layer is a core deep learning building block, and you’ll implement its backward pass from scratch. Given an upstream gradient, compute gradients for the weights, bias, and inputs using the chain rule: $Y = XW + b$.

Requirements

Implement the function

Rules:

• Use the formulas: (dX = dY W^\top), (dW = X^\top dY), (db = \sum_{i=1}^{n} dY_i) (sum over the batch).
• Return each gradient as a NumPy array.
• Don’t use any autograd libraries (no PyTorch, JAX, TensorFlow).
• Use only NumPy and built-in Python libraries.
• Keep it in a single function.

Example

Output: