Chapter 8 - Independence, Basis, and Dimension for Subspaces of (R^n)

Section 8.1 - Subspaces of (R^n)

Example 8.1.5

u={5,-7,11};v={-3,15,9};

uvPlane=ParametricPlot3D[s u+t v,{s,-2,2},{t,-2,2},Mesh-> None, PlotStyle->{Blue,Opacity[0.5]}];

uvPlots=Graphics3D[{Arrowheads[.05],Thickness[.010],Black,Arrow[{{0,0,0},u}],Red,Arrow[{{0,0,0},v}]}];

uvTxtPlots=Graphics3D[{Black,Text["u",{6,-7,12}],Text["v",{-3,15,11}]}];

Show[uvPlane,uvPlots,uvTxtPlots,ViewPoint->{40,-20,-40}]

Example 8.1.13

A={{11,-8,3},{-7,1,-6}};c={{-5},{2}};

RowReduce[Join[A,c,2]]//MatrixForm

RowReduce[Join[A,{{0},{0}},2]]//MatrixForm

S={-t,-t,t};P={-11/45,13/45,0};

T=S+P

ParametricPlot3D[{S,T},{t,-3,3},PlotStyle->{{Red,Thickness[0.005]},{Blue,Thickness[0.005]}},ViewPoint->{2,1,0}]

Example 8.1.14

A={{11,-8,3,2,-1},{-7,1,-6,12,4}};c={{-5},{2}};

RowReduce[Join[A,c,2]]//MatrixForm

RowReduce[Join[A,{{0},{0}},2]]//MatrixForm