ClearAll["Global`*"]

<<Notation`(*load the Notation package*)

Notation[(NotationTemplateTag[(Subscript[a_, i_·])]) ⟺ (NotationTemplateTag[a_[[i_]]])]
Notation[(NotationTemplateTag[(Subscript[a_, i_·j_·])]) ⟺ (NotationTemplateTag[a_[[i_,j_]]])]
(*
In "(Subscript[a_, i_·])", "i_·" (i followed by _ and ·) is the subscript of "a_" (a followed by _),where "·" is the center dot,
use "Ctrl+_" to input the subscript,press "esc" then "." then "esc" to get the center dot "·". 
*)

(*HW1 Question II*)

(∑_(i=1)^3)(Subscript[a, i·i])
(*
(∑_(i=1)^3)(Subscript[a, i·i]) is equivalent to Sum[(Subscript[a, i·i]),{i,1,3}];
In (Subscript[a, i·i]), "i·i" is the subscript of "a";
use "esc"+"sum"+"esc" to get ∑, use ctrl+_ and ctrl+% to input (∑_(i=1)^3).
*)

(∑_(i=1)^3)(∑_(j=1)^3)(Subscript[a, i·j])((Subscript[x, i])+(Subscript[y, j]))

(∑_(i=1)^3)(∑_(j=1)^3)(Subscript[a, i·j])(Subscript[x, i])(Subscript[y, j])

(*HW1 Question III*)

Smat={{1,0,2},{0,1,2},{3,0,3}};
Smat//MatrixForm
avec={1,2,3};
avec//MatrixForm

Sum[Smat[[i,i]],{i,1,3}]
Tr[Smat]

Sum[Smat[[i,j]]Smat[[i,j]],{i,1,3},{j,1,3}]
Tr[Smat.Transpose[Smat]]

Sum[Smat[[i,j]]Smat[[j,i]],{i,1,3},{j,1,3}]
Tr[Smat.Smat]

Sum[avec[[i]]avec[[i]],{i,1,3}]
avec.avec

Sum[Smat[[m,n]]avec[[m]]avec[[n]],{m,1,3},{n,1,3}]
avec.Smat.avec

Format[a[i_,j_]]:=(Subscript[a, i ·j])
Format[a[i_]]:=(Subscript[a, i ])
Format[b[i_]]:=(Subscript[b, i ])

A=Table[a[i,j],{i,1,3},{j,1,3}]
(UnderBar[a])=Table[a[i],{i,1,3}]
(UnderBar[b])=Table[b[i],{i,1,3}]

(UnderBar[a])\[Cross](UnderBar[b])//MatrixForm