Lösungen/Solutions/B1S2Ue/Ex15

\(
L\  = \ 
      Lektion/\ \(Leçon\ \ \ --\)\ Sprechstunde\ /\ heures\ de\ 
        consultation\)W  =  Work / Arbeit / Travail

mF[x_] := MatrixForm[x];

Ueb/Ex 1      W

B = {{1, 0}, {2, 2}}; a = {{1}, {2}}; b = {{2}, {-3}}; 
    v = 2  a - b;

B // mF

a // mF

b // mF

v // mF

B . a // mF

B . b // mF

B . v // mF

p[x_] := 
      Point[{Flatten[x][[1]], Flatten[x][[2]]}]

p[a]

k[x_] := {Flatten[x][[1]], Flatten[x][[2]]}

<< Graphics`Arrow`

ar[z_] := Arrow[{0, 0}, k[z]];
ar[z1_, z2_] := Arrow[k[z1], k[z2]]

Show[
        Graphics[
{PointSize[0.03], 
p[
              a], p[b], p[v], p[B . a], p[B . b], 
            p[B . v], 
ar[a, B . a], ar[b, B . b], 
            ar[v, B . v], 
Line[{{-1, 0}, {5, 0}}], 
            Line[{{0, -1}, {0, 5}}]
}], 
        AspectRatio -> 17/6];

Ueb/Ex 2      W

x = {{x1}, {x2}};  x // mF

A = {{a11, a12}, {a21, a22}};  A // mF

y = x1 {{8}, {7}} + x2 {{2}, {1}};  y // mF

A . x == y

a

Ablesen: / Voir directement:

a11 = 8; a12 = 2; a21 = 7; a22 = 1;  A // mF

b

x = {{5}, {9}};  y = A . x // mF

Ueb/Ex 3      W

R = {{1, 2}, {3, 4}}; T = {{-2, 1}, {1, -2}};

R // mF

T // mF

a

R . T // mF

b

T . R // mF

c

x = {{x1}, {x2}};  x // mF

y = R . x;  y // mF

z = T . y;  z // Simplify // mF

d

(T . R) . x // mF

Ueb/Ex 4      W

M = {{m11, m12}, {m21, m22}};  M // mF

g1 = (M . {{-4}, {0}} == {{1}, {3}})

g2 = (M . {{0}, {-4}} == {{5}, {-1}})

solv = Solve[{g1, g2}, {m11, m12, m21, m22}] // Flatten

M = M /. solv;  M // mF

OP = {{-4}, {-4}};  OQ = M . OP;  OQ //  mF

OS = {{u1}, {u2}};
M . OS == OS

Solve[M . OS == OS, {u1, u2}]

OS = u2 {{-1}, {1}};  OS // mF

SQ = OQ - OS; SQ // mF

SP = OP - OS; SP // mF

SQ == λ SP

solv4 = 
      Solve[SQ == λ SP, {u2, λ}]  // 
        Flatten

OS = u2 {{-1}, {1}} /. solv4;  OS // mF

λ = λ  /. solv4