How does linear algebra facilitate the optimization processes in machine learning models?

Linear algebra lets us represent data, parameters, and predictions as vectors and matrices, so we can use matrix multiplication to compute all predictions at once. The optimization step, like gradient descent, relies on computing gradients, which are expressed as matrix derivatives; this makes updating parameters a simple matrix subtraction. By treating the loss function as a quadratic form, we can solve for the minimum analytically with the normal equations, or approximate it efficiently with iterative methods. For example, in linear regression we compute the weight vector w by solving \(X^T X w = X^T y\) using matrix inversion or decomposition, showing how algebraic operations directly yield the optimal solution. Thus, linear algebra provides the compact notation and computational tools that turn complex optimization into efficient, scalable calculations.