Use the sliders to modify matrix entries. Click and drag locators (small disks) to modify initial points; ALT+Click to add/remove locators.
Manipulate[ListLinePlot[Transpose @ NestList[#.{{a, b}, {c, d}} &, pt, 100],
PlotStyle -> PointSize[Medium], PlotRange -> 5 {{-1, 1}, {-1, 1}},
BaseStyle -> Arrowheads[{0., .05, 0.}], AspectRatio -> Automatic,
PlotLegends -> Placed[LineLegend[Defer /@ pt, LegendLabel -> "{x0,y0}",
LegendFunction -> Panel], Right],
Epilog -> {AbsolutePointSize[10],
{ColorData[97]@#, Point@pt[[#]]} & /@ Range[Length[pt]]},
ImageSize -> 400, Frame -> True] /. Line -> Arrow,
Spacer[10], Spacer[10], Spacer[10],
Grid[{{Item[Labeled[Control@{{a, .8, Style["a", 18]}, 0, 1, Slider,
ImageSize -> Small}, Style[Dynamic[a], 20], Top],
Background -> (Dynamic @ ColorData[{"Rainbow", {-1, 1}}][a])],
Item[Labeled[Control@{{b, .0, Style["b", 18]}, -1, 1, Slider,
ImageSize -> Small}, Style[Dynamic[b], 20], Top],
Background -> (Dynamic @ ColorData[{"Rainbow", {-1, 1}}][b])]},
{Item[Labeled[Control@{{c, .0, Style["c", 18]}, -1, 1, Slider,
ImageSize -> Small}, Style[Dynamic[c], 20], Top],
Background -> (Dynamic@ColorData[{"Rainbow", {-1, 1}}][c])],
Item[Labeled[Control@{{d, .4, Style["d", 18]}, 0, 1, Slider,
ImageSize -> Small}, Style[Dynamic[d], 20], Top],
Background -> (Dynamic@ColorData[{"Rainbow", {-1, 1}}][d])]}},
Alignment -> {Center, Center}, ItemSize -> {15, 15}, Dividers -> All],
{{pt, 3 {{1, 1}, {-1, 1}, {1, -1}}}, Locator,
Appearance -> None, LocatorAutoCreate -> {1, 10}},
Alignment -> Center, ControlPlacement -> Left]
