English: This diagram illustrates the geometric interpretation of least squares for fitting a line y = β₁ + β₂x to three data points. The data vector y in ℝ³ is projected onto the column space of the design matrix X = [1 x], producing the fitted values ŷ = Xβ. The residual vector y − ŷ is orthogonal to col(X). The projected vector y∥ shown in the diagram corresponds to the fitted values ŷ, and y⊥ = y − ŷ is the residual vector. Each axis represents the values at the three data points (y₁, y₂, y₃), so each point in ℝ³ corresponds to a vector of data values. The projection onto col(X) corresponds to selecting the line whose predicted values best approximate the data in the least squares sense.