In machine learning, the performance of a model often improves with the optimization of its parameters through linear algebra techniques. What is the underlying reason why matrix operations are so critical in this optimization process?

In machine learning, the model’s parameters are usually stored as vectors or matrices, and the data that the model processes is also represented in matrix form, so that all calculations can be expressed as matrix operations. Linear algebra lets us combine many operations—such as multiplying a weight matrix by an input vector—into a single, highly efficient matrix multiplication, which modern hardware can accelerate with SIMD or GPU kernels. Because the loss function is a smooth function of these parameters, gradient‑based optimization requires repeatedly computing gradients, which are themselves linear combinations of matrix products. Thus, matrix operations provide the mathematical backbone that lets us compute gradients, propagate errors, and update parameters all at once, making the entire training loop fast, scalable, and numerically stable. For example, a single hidden layer in a neural network applies a weight matrix to an input vector to produce the next layer’s activations, and the back‑propagation step uses the transpose of that matrix to update the weights efficiently.