If vectors vec$(u)=(1,1,-1)$ and vec$(v)=(-2,5,1)$ are given, calculate i$){}_{ij}{u}_i{v}_j=?$

ii$)a_{ij}=u_i{v}_j$

iii$)p_i=a_{ij}{u}_j$

a$)$ Show that the vectors vec$(v{)}_1=(1,2,0),$vec$(v{)}_2=(1,3,2),$ and vec$(v{)}_3=(2,0,-1)$ form a linearly independent set. Express vec$(u)=(1,-1,3)$ as a linear combination of the three vectors vec$(v{)}_1,$vec$(v{)}_2,$ and vec$(v{)}_3.$

b$)$ If the linearly independent set of vectors vec$(v{)}_1=(1,2,0),$vec$(v{)}_2=(1,3,2)$ and vec$(v{)}_3=(2,0,-1)$ are given, obtain the orthonormal set vec$(u{)}_1,$vec$(u{)}_2,$vec$(u{)}_3.$