Ordinary least squares
Projection

A geometrical interpretation of these equations is that the vector of residuals, y − X β ^ {\displaystyle \mathbf {y} -X{\hat {\boldsymbol {\beta }}}} is orthogonal to the column space of X, since the dot product (y − X β ^) ⋅ X v {\displaystyle (\mathbf {y} -\mathbf {X} {\hat {\boldsymbol {\beta }}})\cdot \mathbf {X} \mathbf {v} } is equal to zero for any conformal vector, v. This means that y − X β ^ {\displaystyle \mathbf {y} -\mathbf {X} {\boldsymbol {\hat {\beta }}}} is the shortest of all possible vectors y − X β {\displaystyle \mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}}, that is, the variance of the residuals is the minimum possible. This is illustrated at the right.
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