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Linear models for regression I [2 P]

The pseudo-inverse $ (\Phi^T \Phi)^{-1} \Phi^T$ of the design matrix $ \Phi$ can be used to directly calculate the optimal weights of a linear model.

Prove: If the columns of the $ i \times j$ design matrix $ \Phi$ are linearly independent, then the $ j \times j$ matrix $ \Phi^T \Phi$ is invertible.

Hint: Prove first that if $ \Phi^T \Phi  {\bf a} = {\bf0}$ for an arbitrary vector $ {\bf a} \in \mathbb{R}^n$ , then $ \Vert{\Phi}  {\bf a} \Vert = 0$ .



Haeusler Stefan 2010-01-19