In terms of \(M\), recall that \(L(V)\) is spanned by the columns of \(M\). (In the standard basis, the columns of \(M\) are \(Me_{1}\), \(\ldots\), \(Me_{n}\).) Then \(v^{\perp}\) must be perpendicul...In terms of \(M\), recall that \(L(V)\) is spanned by the columns of \(M\). (In the standard basis, the columns of \(M\) are \(Me_{1}\), \(\ldots\), \(Me_{n}\).) Then \(v^{\perp}\) must be perpendicular to the columns of \(M\). \(\textit{i.e.}\), \(M^{T}(V-MX)=0\), or