Practical Linear Algebra: A Geometry Toolbox, Third edition
Gerald Farin & Dianne Hansford
CRC Press, Taylor & Francis Group, An A K Peters Book, 2013

Linear Systems viewed as a line intersection problem

These examples come from Chapter 5 in PLA.
1. unique solution, 
2. inconsistent linear system, 
3. overdetermined linear system.
4. homogeneous sys with trivial solution only
5. homogeneous sys with nontrivial solutions

In the book, mostly we  take a “column view” -- or coordinate system transformation view. This demonstrates that a “row view” is also possible, and the classification of the systems is identical.

SetDirectory[NotebookDirectory[]];
SetOptions[Graphics,BaseStyle->{FontFamily->"Helvetica",FontSize->10}];

System #1 : unique solution
  2(Subscript[u, 1]) + 4(Subscript[u, 2])= 4
  (Subscript[u, 1]) + 6(Subscript[u, 2]) = 4
  
  (({{2, 4}, {1, 6}}))(({{(Subscript[u, 1])}, {(Subscript[u, 2])}})) = (({{4}, {4}}))

A1 = {{2,4},{1,6}};
b1 = {4,4};
u1 = LinearSolve[A1,b1];
Print["Example 1: Intersection point =",u1];

(* Select endpoints of lines for good graphics *)
ex1l1 = {{-3,10/4},{4,-1}};
ex1l2 = {{-3,7/6},{4,0}};

plotEx1 = Graphics[{Thick,Black, Line[ex1l1], Gray, Line[ex1l2], Black, PointSize[Large],Point[u1]}, Axes->True, Ticks->{{-3,-2,-1,0,1,2,3},{-1,0,1,2}}]

System #2 : inconsistent

  2(Subscript[u, 1]) + (Subscript[u, 2]) = 1
  4(Subscript[u, 1]) + 2(Subscript[u, 2]) = 1
    
  (({{2, 1}, {4, 2}}))(({{(Subscript[u, 1])}, {(Subscript[u, 2])}})) = (({{1}, {1}}))

(*
ex2l1 = {{-1,3},{1,-1}};
ex2l2 = {{-1,5/2},{1,-3/2}};
*)
ex2l1 = {{-1/2,2},{1/2,0}};
ex2l2 = {{-1/2,3/2},{1/2,-1/2}};

plotEx2 = Graphics[{Thick,Black, Line[ex2l1], Gray, Line[ex2l2], Black}, Axes->True,Ticks->{{-1/2,0,1/2},{-1,0,1,2,3}}]

System #2 : underdetermined

2(Subscript[u, 1]) + (Subscript[u, 2]) = 3
4(Subscript[u, 1]) + 2(Subscript[u, 2]) = 6

  (({{2, 1}, {4, 2}}))(({{(Subscript[u, 1])}, {(Subscript[u, 2])}})) = (({{3}, {6}}))

(* ex3l1 = {{-2,7},{3,-3}}; *)
ex3l1 = {{-1,5},{2,-1}};
plotEx3 = Graphics[{Thick,Black, Line[ex3l1], Black}, Axes->True, Ticks->{{-2,-1,0,1,2},{-2,-1,0,1,2,3,4,5,6}}]

pic = GraphicsRow[{plotEx1, plotEx2, plotEx3},Alignment->Center, ImageSize->Full]
Export["LinearSysLines.eps", pic];

Homogeneous systems : trivial solution and nontrivial solution

(* Homogeneous - trivial solution *)
hex1l1 = {{-2,1},{2,-1}};
hex1l2 = {{-2,4},{2,-4}};
plothex1 = Graphics[{Thick, Black, Line[hex1l1], Gray, Line[hex1l2]}, Axes->True];


(* Homogeneous -- nontrivial soln -- identical lines *)
hex2l1 = {{-3,3/2},{2,-1}};
plothex2 = Graphics[{Thick, Black, Line[hex2l1]}, Axes->True];

pic2 =GraphicsRow[{plothex1, plothex2},Alignment->Center, ImageSize->Full]
Export["HomoSysLines.eps", pic2];