## Linear Algebra Question (projection, transformation)

1. Consider subspace W = Span $

\begin{bmatrix} 1 & 2 & 0 \\ 1 & 0 & 6 \\ 1 & 2 & -2 \\ -1 & 0 & -4\end{bmatrix}

$

a. Find the orthonormal basis S for W

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

b. Find the Orthogonal Projection of V=
$
\begin{bmatrix} a\\b\\c\\d \end{bmatrix}
$

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 $
\begin{bmatrix} 1\\0\\0\end{bmatrix} ,
\begin{bmatrix} 1\\1\\0\end{bmatrix} ,
\begin{bmatrix} 1\\1\\1\end{bmatrix}
$
of $R_3$ and
$
\begin{bmatrix} 1\\1\end{bmatrix} ,
\begin{bmatrix} 1\\2\end{bmatrix}
$
of $R_2$

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

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