For the set of vectors

vec$(v_1)=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right],$vec$(v_2)=\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right],$vec$(v_3)=\left[\begin{array}{c}1\\ 1\\ 2\end{array}\right]$

a$)$ Show the vectors vec$(v_1),$vec$(v_2),$vec$(v_3)$ are linear independent.

b$)$ Find vec$(e_1),$ which is the normalized vector of vec$(v_1).$

c$)$ Find vec$(e_2),$ which is the normalized vector of vec$(u_2),$ and vec$(u_2)=$vec$(v_2)-pro{j}_{\text{v}\text{e}\text{c}(v_1)}$vec$(v_2)$

d$)$ Find vec$(e_3),$ which is the normalized vector of vec$(u_3),$ and

vec$(u_3)=$vec$(v_3)-pro{j}_{\frac{?}{\text{b}\text{a}\text{r}}(v_1)}$vec$(v_3)-pro{j}_{\frac{?}{\text{b}\text{a}\text{r}}(u_2)}$vec$(v_3).$

e$)$ Show the vector set, $\{vec(e_1),$vec$(e_2),$vec$(e_3)\},$ is orthonormal.