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

Linear Maps in 3D
Linear maps applied to 3D objects 
See Figures 9.2-4

Define some 3D parametric objects

torus3D[u_,v_] :={(3+Cos[v])Cos[u],(3+Cos[v])Sin[u],Sin[v]};
parab3D [r_,t_] := {r*Cos[t], r*Sin[t], r^2};

Uniform scale

mat = {{1/3, 0, 0},{0, 1/3,0},{0,0,1/3}};

Print["Uniform scale ",MatrixForm[mat]," applied to the torus",MatrixForm[torus3D[uu,vv]]];

ParametricPlot3D[{torus3D[u,v],mat.torus3D[u,v]},{u,0,2Pi},{v,0,2Pi},Lighting->"Neutral",PlotStyle->Specularity[White,100], Boxed->False, Axes->False]

Non - uniform Scale

mat = {{1/3, 0, 0},{0, 1,0},{0,0,3}};
Print["nonuniform scale ",MatrixForm[mat]," applied to parametric torus ",MatrixForm[torus3D[uu,vv]]];

ParametricPlot3D[{torus3D[u,v],mat.torus3D[u,v]},{u,0,2Pi},{v,0,2Pi},Lighting->"Neutral",PlotStyle->Specularity[White,100], Boxed->False, Axes->False]

Shear

mat = {{1, 0, 1},{0, 1,1},{0,0,1}};
Print["nonuniform scale ",MatrixForm[mat]," applied to paraboloid ",MatrixForm[parab3D[rr,tt]]];

ParametricPlot3D[{parab3D[r,t],mat.parab3D[r,t]},{t,0,2Pi},{r,0,2},PlotRange->Automatic, Lighting->"Neutral",Boxed->False, Axes->False ]
(* {(r*Cos[t]) +(r^2)*1, (r*Sin[t])+(r^2)*1, r^2}} *)

Just playing

mat = {{1/2, 0, 4},{0, 2,1},{0,0,4}};
Print["Linear map ",MatrixForm[mat]," applied to parametric torus ",MatrixForm[torus3D[uu,vv]]];

ParametricPlot3D[{torus3D[u,v],mat.torus3D[u,v]},{u,0,2Pi},{v,0,2Pi},Lighting->"Neutral",PlotStyle->Specularity[White,100], Boxed->False, Axes->False]