and the only norm is the
1. Let
x = \begin{bmatrix} 1\\1\\2 \end{bmatrix} and y = \begin{bmatrix} -2\\1\\2 \end{bmatrix}.
Find the vector projection p of x onto y.
2. For y in R<sup>n</sup>, let A be the (n Ć 1)-matrix consisting of the column vector y. Show
that for any x in R<sup>n</sup>,
A(A<sup>T</sup>A)<sup>-1</sup>A<sup>T</sup>x = \frac{(x \cdot y)}{(y \cdot y)}y.
3. A square (n Ć n)-matrix P is called a projection matrix if P<sup>2</sup> = P and P<sup>T</sup> = P.
For x in R<sup>n</sup>, let p = Px. Show x - p is in Col(P)<sup>?</sup>.
4. Let S be the subspace in R<sup>3</sup> spanned by {(1, 1, 2)<sup>T</sup>, (-2, 1, 2)<sup>T</sup>}. Find the matrix
P that projects a vector onto S.
5. Repeat problem (4) with the orthonormal basis
\begin{Bmatrix} v_1 = \begin{pmatrix} 1/\sqrt{6}, 1/\sqrt{6}, 2/\sqrt{6} \end{pmatrix}<sup>T</sup>, v_2 = \begin{pmatrix} -5/\sqrt{30}, 1/\sqrt{30}, 2/\sqrt{30} \end{pmatrix}<sup>T} \end{Bmatrix}
