Koordinaten in Vektorräumen

Beispiel 5.21

Man bestimme die Koordinaten des Vektors a = (({{x}, {y}, {z}})) ϵ (ℝ^3) bezüglich der Basis b[1] = (({{1}, {1}, {0}})), b[2] = (({{0}, {-1}, {0}})), b[3] = (({{0}, {0}, {3}})) .

b[1]=(({{1}, {1}, {0}}));b[2]=(({{0}, {-1}, {0}}));b[3]=(({{0}, {0}, {3}}));a=(({{x}, {y}, {z}}));

Wir zeigen zuerst, dass wir eine Basis haben:

Solve[(∑_(k=1)^3)λ[k] b[k]==0,Table[λ[k],{k,1,3}]]

Nun soll die Linearkombination gleich a sein

sol=Solve[(∑_(k=1)^3)λ[k] b[k]==a,Table[λ[k],{k,1,3}]]

Übergangsmatrix

B=Transpose[Join[Transpose[b[1]],Transpose[b[2]],Transpose[b[3]]]]

Inverse[B].a

Beispiel 5.23

b[1]=(({{1}, {2}, {-1}}));b[2]=(({{2}, {0}, {3}}));b[3]=(({{1}, {2}, {0}}));

M1=Transpose[Join[Transpose[b[1]],Transpose[b[2]],Transpose[b[3]]]]

4Inverse[M1]

btilde[1]=(({{2}, {0}, {1}}));btilde[2]=(({{0}, {3}, {0}}));btilde[3]=(({{1}, {2}, {1}}));

M2=Transpose[Join[Transpose[btilde[1]],Transpose[btilde[2]],Transpose[btilde[3]]]]

Btilde=Inverse[M1].M2

Table[btilde[j]==Sum[βtilde[k,j]b[k],{k,1,3}],{j,1,3}]

Solve[Table[btilde[j]==Sum[βtilde[k,j]b[k],{k,1,3}],{j,1,3}],Flatten[Table[βtilde[k,j],{k,1,3},{j,1,3}]]]

Solve[Table[b[k]==Sum[β[k,j]btilde[j],{j,1,3}],{k,1,3}],Flatten[Table[β[k,j],{j,1,3},{k,1,3}]]]

Inverse[M2].M1

B=Inverse[Btilde]