1. Consider subspace W = Span <br /> <br /> \begin{bmatrix} 1 & 2 & 0 \\ 1 & 0 & 6 \\ 1 & 2 & -2 \\ -1 & 0 & -4\end{bmatrix}<br /> <br />

a. Find the orthonormal basis S for W

Using Gram-schmidt process: S= <br /> u_1\begin{bmatrix} 1/2\\1/2\\1/2\\-1/2\end{bmatrix} ,<br /> u_2\begin{bmatrix} 1/2\\-1/2\\1/2\\1/2\end{bmatrix} ,<br /> u_3\begin{bmatrix} 1/2\\1/2\\-1/2\\1/2\end{bmatrix}<br />

b. Find the Orthogonal Projection of V=
<br /> \begin{bmatrix} a\\b\\c\\d \end{bmatrix}<br />
onto w and w complement?

Im not sure one how to do it? Any help would be appreciated.

2.Let R_3 \longrightarrow R_2 be defined by

L( \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}) = \begin{bmatrix}x_1+2x_2+2x_3\\x_1+2x_x+x_3 \end{bmatrix}

Find the matrix representing L with respect to the bases <br /> \begin{bmatrix} 1\\0\\0\end{bmatrix} ,<br /> \begin{bmatrix} 1\\1\\0\end{bmatrix} ,<br /> \begin{bmatrix} 1\\1\\1\end{bmatrix}<br /> of R_3 and
<br /> \begin{bmatrix} 1\\1\end{bmatrix} ,<br /> \begin{bmatrix} 1\\2\end{bmatrix} <br /> of R_2

I can find the bases with respect to R_3 which is L= <br /> \begin{bmatrix} 1 & 3 & 5\\1 & 3 &4\end{bmatrix} <br /> but im unsure on how to approach the R_2 one.

Any help on these 2 problems will be appreciated. Thanks in advance