Linear regression isn't just about drawing a line through scattered points. Vector projections reveal the core mechanism: finding the closest vector to your target within the space spanned by your features. This shift from feature space to column space, where each house becomes an axis, reframes the problem geometrically. The article demonstrates how minimizing residuals translates to projecting the price vector onto the size vector's direction, explaining why calculus-heavy derivations become messy in high dimensions. Column space becomes the crucial arena where regression happens, simplifying the understanding of how a single weight scales an independent variable to approximate the dependent variable.