14Nov2023:

(ip-select t)
  "Using COLVEC-inner-product."

(setq u (mat-make-colvec 1 1 0))
    [  1  ]
    [  1  ]
    [  0  ]

(setq v (mat-make-colvec  0 1 2))


(setq b (mat-make-colvec  -1 2 3))


;; Let L := Spn(u,v) and let
;;       bOnL 
;; be the ortho-projection of b onto plane L.
;; Is  bOnL   
;; just the sum of projections on u and v separately?

;; No! vectors u and v interact, as they are not a perpendicular pair.
;; So lets rewrite L as the span of a perpendicular pair.


(proj v u)
    [  1/2  ]
    [  1/2  ]  ;  1/2
    [    0  ]


;; So a vector in L that is perpendicular to u is:
(setq vHat (mat-sub v (proj v u)))
    [  -1/2  ]
    [   1/2  ]
    [     2  ]

(ip vHat u) => 0  ;; Yup.

;; Making vHat nicer.
(setq vHat (mat-scal-mult 2 vHat))
    [  -1  ]
    [   1  ]
    [   4  ]


;;;;;;;;;;;;;;;;
;; Now for the b projections.
;;;;;;;;;;;;;;;;
(setq bProjOnu (proj b u))
    [  1/2  ]
    [  1/2  ]
    [    0  ]


(setq bProjOnvHat (proj b vHat))
    [  -5/6  ]
    [   5/6  ]
    [  10/3  ]


;; Drum roll, please...
(setq bOnL  (mat-add bProjOnu bProjOnvHat))
    [  -1/3  ]
    [   4/3  ]
    [  10/3  ]

;;;;;;;;;;;;;;;;
;;Checking.  First compute the supposed
;; orthogonal-to-L vector:
;;;;;;;;;;;;;;;;
(setq bDiff (mat-sub b bOnL))
    [  -2/3  ]
    [   2/3  ]
    [  -1/3  ]

(ip bDiff u) => 0
(ip bDiff v) => 0

;; Now  let's jeck that bOnL is indeed 
;; in the L plane.
(setq M (mat-Horiz-concat u v bOnL))
    [  1  0  -1/3  ]
    [  1  1   4/3  ]
    [  0  2  10/3  ]

(rref-mtab-beforecol M)

  JK: Found 2 pivots before the third column.


       c0   c1   c2      Row operations
   |------------------|
r0 |    1    0  -1/3  |    1    0    0
r1 |    0    1   5/3  |   -1    1    0
r2 |    0    0    0   |    2   -2    1
   |------------------|


;; Yup; The rightmost column is in the span
;; of the two columns to its left.