Review for quiz 3

We have our third quiz next Wednesday, November 19th. Here’s our review sheet for that quiz.

The problems

1. Write down the definition of each of the following:

   1. Orthogonal vectors Def 6.1.7
   2. Orthonormal set of vectors Def 6.3.6
   3. Orthogonal matrix Def 6.3.22
   4. Orthogonal projection Definition 6.3.12

2. Let \(\mathbf u\) and \(\mathbf v\) denote the 3D vectors \[ \mathbf{u} = \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix} \: \text{ and } \: \mathbf{v} = \begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}. \]

   1. Verify that \(\mathbf u\) and \(\mathbf v\) are orthogonal.
   2. Find the matrix \(P\) that projects orthogonally onto the subspace spanned by \(\mathbf u\) and \(\mathbf v\).

   I guess you might consider using the technique outlined in Proposition 6.3.16

3. Use the normal equations to find the least squares regression line for the points \[ \{(-1,-1), (0,1), (2,0)\}. \]

Your questions and solutions

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