A data scientist is building a machine learning model to predict house prices based on various features such as size, location, and number of bedrooms. Which linear algebra principle is most critical for optimizing the model's parameters during training?

The most critical linear‑algebra idea is that the model’s parameters are found by solving a system of linear equations, which is done with matrix multiplication and inversion. In ordinary least squares, you compute \(X^TX\) and then its inverse to get \(\theta=(X^TX)^{-1}X^Ty\). This uses the dot product to build the matrices and the inverse to solve for the best‑fit weights. For example, if you have three features, you form a 3×3 matrix from the data, invert it, and multiply by the target vector to get the price‑prediction coefficients. This matrix‑based solution is the core of training a linear regression model.