Problem 1

Block[{a1, a2,a, g, g1, g2},
a1 = {2, 0} ;
a2 = {1.75, 2.6}/2 ;

(* verify projection measurements against measurement of length and angle *)
(*{Norm[a2],
ArcTan[a2[[2]]/a2[[1]]] / °}*)

a = { a1, a2} ;
g = 2 Pi Inverse[a] ;
g1=Part[g,All,1];g2=Part[g,All,2];

(* check "orthonormality *)
(*{a1 . g1, a2 . g2, a1 . g2, a2 . g1 } / (2 Pi), a . g/ (2 Pi) // MatrixForm }*)

{
(*, a // MatrixForm *)
 g // MatrixForm
 , g1 // MatrixForm
, g2 // MatrixForm
, g1 /(2 Pi)// MatrixForm
, g2 /(2 Pi)// MatrixForm
} 
]

Problem 2

Block[{a},
a = { {1, 1,0}, {1, 0, 1}, {0, 1, 1}} ;
2 Inverse[a] // MatrixForm
]