;; Given a list G1, G2, G3, ...   of vectors, we 
; compute an orthogonal system
;              B1, B2, B3, ...  with the same span.
;
;   In particular if G1, G2, G3, ...   was a basis, 
; then the B-system is an orthogonal basis.
;
;   We could normalize the B-vectors, to get
; an ortho-normal basis  N1, N2, N3, ...
; The dot-product w.r.t that ortho-normal basis will
; equal the abstract inner-product that we started with.

% (use-ring Rational-ring)

% (setq G1 (mat-make-initseq 3 1 '(1 0 0)))
    [  1  ]
    [  0  ]
    [  0  ]

% (setq G2 (mat-make-initseq 3 1 '(2 -3 0)))
    [   2  ]
    [  -3  ]
    [   0  ]

% (setq G3 (mat-make-initseq 3 1 '(-1 2 4)))
    [  -1  ]
    [   2  ]
    [   4  ]

;; Computing an orthogonal {B1, B2, B3} with the
;; same span as {G1, G2, G3}.
% (setq B1 G1)
    [  1  ]
    [  0  ]
    [  0  ]

% (setq ratio (/ (ip B1 G2) (ip B1 B1))) -> 2

% (setq ProjG2onB1 (mat-scal-mult ratio B1))
    [  2  ]
    [  0  ]
    [  0  ]

% (setq B2 (mat-sub G2 ProjG2onB1))  ;;This is Orth_B1(G2).
    [   0  ]
    [  -3  ]
    [   0  ]

% (ip B1 B2) -> 0  ; Checking the B-vectors are orthogonal.

;; Computing B3.
% (setq ProjG3onB1
   (mat-scal-mult  (/ (ip B1 G3) (ip B1 B1))  B1) )

    [ -1 ]
    [  0 ]
    [  0 ]


% (setq ProjG3onB2
    (mat-scal-mult  (/ (ip B2 G3) (ip B2 B2))  B2) )

    [ 0 ]
    [ 2 ]
    [ 0 ]

% (setq B3  (mat-sub G3 (mat-add ProjG3onB1 ProjG3onB2)))
    [ 0 ]
    [ 0 ]
    [ 4 ]

Easily

    [ -1 ]     [ 0 ]     [ 0 ]
    [  0 ] ,   [ 2 ] ,   [ 0 ]
    [  0 ]     [ 0 ]     [ 4 ]

is an orthogonal basis for |R^3.

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