v1={1,2};
v2={-2,-3};
v3={5,-2};

x varies from -2 to 5
y varies from -3 to 2

Show[Graphics[{Raster[Table[If[x<0&&y<0,{0,0,0},{x,y,Sin[x y]}], {x,-2,5,.1},{y,-3,2,.1}]]}],Axes->True]

T={{v1[[1]]-v3[[1]],v2[[1]]-v3[[1]]},{v1[[2]]-v3[[2]],v2[[2]]-v3[[2]]}};
invT=Inverse[T];

temp=invT.({x,y}-v3);
barycoord=AppendTo[temp,1-temp[[1]]-temp[[2]]];

Show[Graphics[{Raster[Table[If[
0≤barycoord[[1]]≤1
&&
0≤barycoord[[2]]≤1
&&
0≤barycoord[[3]]≤1
,{0,0,0},barycoord], {x,-2,5,.1},{y,-3,2,.1}]]}],Axes->True]