$Post:=If[MatrixQ[#],MatrixForm[#],#]&
<<LinearAlgebra`Master`

Exercise 2.6.1

Do[v[i]=Table[1/(i+j),{j,3}],{i,3}]
B=Array[v[#]&,3]

RowReduce[B]

The v's are linearly independent.

G=GramSchmidt[B]

Check:

Table[G[[i]].G[[j]],{i,3},{j,3}]

Exercise 2.6.2

Do[v[i]=Table[i/(i+j),{j,8}],{i,6}]
B=Array[v[#]&,6]

RowReduce[B]

The v's are linearly independent.

G=GramSchmidt[B]

Check:

Table[G[[i]].G[[j]],{i,6},{j,6}]

Exercise 2.6.3

Do[G[[i]]=Normalize[G[[i]]],{i,6}]
G

Obviously from the previous exercise, the GramSchmidt
procedure of Mathematica already normalizes the basis.
But if this were not the case or if we were using a
procedure of our own, the above would do (non-redundantly)
the normalization.

Exercise 2.6.4

Do[v[i]=Table[1/(i+j),{j,8}],{i,6}]
B=Array[v[#]&,6]

iprod[u_,v_]:=Sum[u[[i]]v[[i]]i,{i,8}]
proj[v_][u_]:=iprod[v,u]/iprod[v,v]v
Do[w[i]=v[i]-Sum[proj[w[j]][v[i]],{j,i-1}],{i,6}]
Array[w[#]&,6]

Check:

Table[Normalize[Table[iprod[w[i],w[j]],{j,6}]],{i,6}] \
  //Together