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

Linear combinations 3D
An example of a linear space is  all 3D vectors v=(({{(Subscript[x, 1])}, {(Subscript[x, 2])}, {(Subscript[x, 3])}})) .
Linear combinations are defined by
s1(Subscript[v, 1])+s2(Subscript[v, 2])+s3(Subscript[v, 3]);siϵ ℝ. 
The zero vector now is 0=(({{0}, {0}, {0}})).
A matrix is defined by(({{(Subscript[a, 11]), (Subscript[a, 12]), (Subscript[a, 13])}, {(Subscript[a, 21]), (Subscript[a, 22]), (Subscript[a, 23])}, {(Subscript[a, 31]), (Subscript[a, 32]), (Subscript[a, 33])}})).

The gray triangle in the following cell defines the plane formed by 
a,b,c (in red, green, blue, resp.).
The result of the linear coombinations is shown as a 3D tube vector.

aa = {1,0,0};(*  Red Arrow *)
bb = {0,1,0}; (*  Green Arrow *)
cc = {0,0,1};  (*  Blue Arrow *)

arrowaa={Red,Arrowheads[Large],Arrow[{{0,0,0},aa}]};
arrowbb ={Green,Arrowheads[Large],Arrow[{{0,0,0},bb}]};
arrowcc ={Blue,Arrowheads[Large],Arrow[{{0,0,0},cc}]};

(* plotting area range: -range to + range in each coordinate *)
range = 2.1;
(* r,s,t parameter extents *)
xtnts = 2;

Manipulate[
Graphics3D[{ {Arrowheads[.06],
Arrow[Tube[{{0,0,0},r*aa+s*bb+t*cc},.008]]},
Thick,arrowaa, arrowbb, arrowcc, 
LightGray,Opacity[0.8],Polygon[{aa,bb,cc}]},
Boxed->False,PlotRange->{{-range,range},{-range,range},{-range,range}}],{{r,1},-xtnts,xtnts},{{s,1},-xtnts,xtnts},{{t,1},-xtnts,xtnts},
FrameLabel->{" "," ","Linear combination: v = r*red + s*green + t*blue"," "}]