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

Linear Dependence and Determinants

Two vectors a and b are called linearly dependent if one can be expressed in terms of the other:
	b=ra.
Else, they are linearly independent.

The determinant of two vectors measures the area of the parallelogram spanned by them. If it is zero, then the vectors are linearly dependent. The area may also be negative. In the following cell, the determinant is displayed in a box on top of the graphic.

(* Fixed vector -- in black *)
a1 = {1,0}; 

(* graphics for a1 *)
arrowa1={Thick,Arrowheads[Large],Arrow[{{0,0},a1}]};
arrowplot=Graphics[arrowa];

Manipulate[
b1={s,t};  (*in red *)
arrowb1={Thick,Arrowheads[Large],Arrow[{{0,0},b1}]};

Graphics[{Green,Polygon[{{0,0},{1,0}, {1,0}+b1,b1}],Black,arrowa1,Red,arrowb1},PlotLabel->Style[Framed[Det[{{1,0},b1}]],20],Axes->True,PlotRange->{{-2,2},{-1,1}}],

{{s,0.5},-1,1},{{t,0.5},-1,1},
FrameLabel->{" "," ","Determinant: area of the parallelogram"," "}]