Find the orthogonal projection of $\vec{v}=\left[\begin{array}{c}22 \\ -26 \\ 8 \\ -16\end{array}\right]$ onto the subspace $V$ of $\mathbb{R}^{4}$ spanned by $\left[\begin{array}{c}1 \\ -5 \\ -3 \\ -1\end{array}\right],\left[\begin{array}{c}5 \\ -1 \\ 3 \\ 1\end{array}\right]$, and $\left[\begin{array}{c}-1 \\ -1 \\ 3 \\ -5\end{array}\right]$.
(Note that these three vectors form an orthogonal set.)
\[
\operatorname{proj}_{V}(\vec{v})=\left[\begin{array}{l}
\square \\
\square \\
\square \\
\square
\end{array}\right]
\]

Step 1 ：Define the vector \( \vec{v} = \left[\begin{array}{c}22 \ -26 \ 8 \ -16\end{array}\right] \) and the basis vectors for the subspace \( V \) as \( \vec{u_1} = \left[\begin{array}{c}1 \ -5 \ -3 \ -1\end{array}\right] \), \( \vec{u_2} = \left[\begin{array}{c}5 \ -1 \ 3 \ 1\end{array}\right] \), and \( \vec{u_3} = \left[\begin{array}{c}-1 \ -1 \ 3 \ -5\end{array}\right] \).

Step 2 ：Calculate the orthogonal projection of \( \vec{v} \) onto each basis vector. The projection of \( \vec{v} \) onto \( \vec{u_1} \) is \( \vec{proj_{u_1}v} = \frac{\vec{v} \cdot \vec{u_1}}{\vec{u_1} \cdot \vec{u_1}} \vec{u_1} = \left[\begin{array}{c}4 \ -20 \ -12 \ -4\end{array}\right] \). Similarly, the projection of \( \vec{v} \) onto \( \vec{u_2} \) is \( \vec{proj_{u_2}v} = \frac{\vec{v} \cdot \vec{u_2}}{\vec{u_2} \cdot \vec{u_2}} \vec{u_2} = \left[\begin{array}{c}20 \ -4 \ 12 \ 4\end{array}\right] \), and the projection of \( \vec{v} \) onto \( \vec{u_3} \) is \( \vec{proj_{u_3}v} = \frac{\vec{v} \cdot \vec{u_3}}{\vec{u_3} \cdot \vec{u_3}} \vec{u_3} = \left[\begin{array}{c}-3 \ -3 \ 9 \ -15\end{array}\right] \).

Step 3 ：Sum the projections to find the orthogonal projection of \( \vec{v} \) onto \( V \). The orthogonal projection of \( \vec{v} \) onto \( V \) is \( \vec{proj_Vv} = \vec{proj_{u_1}v} + \vec{proj_{u_2}v} + \vec{proj_{u_3}v} = \left[\begin{array}{c}21 \ -27 \ 9 \ -15\end{array}\right] \).

Step 4 ：\(\boxed{\text{Final Answer: The orthogonal projection of } \vec{v} \text{ onto the subspace } V \text{ is } \vec{proj_Vv} = \left[\begin{array}{c}21 \ -27 \ 9 \ -15\end{array}\right]}\)