3.9 Design Matrix of Less Than Full Rank: 3.9.1 Least Squares Estimation

Suppose that $\mathbf{X}$ does not have full rank, and let $\widehat{\mathbf{\beta }}_{i}(i = 1,2)$ be any two solutions of the normal equations. Show directly that

If the columns of $\mathbf{X}$ are linearly dependent, prove that there is no matrix $\mathbf{C}$ such that $\mathbf{C Y}$ is an unbiased estimate of $\mathbf{\beta }$.