For the linear estimator β ~ ​ =Cy to be unbiased, its expected value must equal the true parameter vector β, no matter what the value of β is. The condition CX=I is precisely what guarantees this.

Var(Cy∣X)=CVar(y∣X)C′

The matrix DD ′ is positive semidefinite, meaning its diagonal elements are greater than or equal to zero.

Var( β ^ ​ ∣X)=E[( β ^ ​ −β)( β ^ ​ −β) ′ ∣X]

If the columns of X are linearly dependent, there must exist a non-zero k x 1 vector c such that: Xc=0 This equation is the mathematical definition of linear dependence. It states that there's a non-zero combination of the columns of X (with weights from c) that results in the zero vector.