Q2

Q2.pdf

1

$$v\in V\cap W\to v\perp v\to |v|=0$$

2

$$\begin{gathered} A=\left[\begin{array}{ccc}1& 2& 1\\ 2& 4& 3\\ 3& 6& 4\end{array}\right]\to \left[\begin{array}{ccc}1& 2& 1\\ 0& 0& 1\\ 0& 0& 0\end{array}\right] \\ x\perp \{\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\};x=\left[\begin{array}{c}2\\ -1\\ 0\end{array}\right] \\ y\perp \{\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]\};y=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right] \\ z\perp \{\left[\begin{array}{c}2\\ -1\\ 0\end{array}\right]\};z=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right] \end{gathered}$$

4

$$\begin{gathered} \left[\begin{array}{cccc}1& 1& 1& 0\\ 1& 0& -1& 0\end{array}\right]\to \left[\begin{array}{cccc}1& 0& -1& 0\\ 0& 1& -2& 0\end{array}\right] \\ \{\begin{array}{l}{x}_1-x_3=0\\ x_2-2x_3=0\end{array} \\ a=x_3\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right] \\ P=\frac{a{a}^T}{a^{T}a} \end{gathered}$$

5

$$\begin{gathered} tD=b \\ \left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]\left[\begin{array}{c}D\end{array}\right]=\left[\begin{array}{c}1\\ 7\end{array}\right] \\ \hat{D}=(t^{T}t{)}^{-1}{t}^{T}b \end{gathered}$$