Aufgabe 62

M = {{1, 1, 2, 0}, {2, 1, 3, 1}, {1, 0, 1, 1}}

null = NullSpace [ M]

Dim(Kern(f))=2

v[1] = 
      M . {{x}, {y}, {z}, {u}} /. {x -> 1, y -> 0, z -> 0, 
          u -> 0}

v[2] = 
      M . {{x}, {y}, {z}, {u}} /. {x -> 0, y -> 1, z -> 0, 
          u -> 0}

v[3] = 
      M . {{x}, {y}, {z}, {u}} /. {x -> 0, y -> 0, z -> 1, 
          u -> 0}

v[4] = 
      M . {{x}, {y}, {z}, {u}} /. {x -> 0, y -> 0, z -> 0, 
          u -> 1}

Frage: Welche dieser Vektoren sind linear unabhängig?

Solve[(∑_(k = 1)^4) λ[k]*v[k] == 0, 
      Table[λ[k], {k, 1, 4}]]

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

Solve[(∑_(k = 1)^2) λ[k]*v[k] == 0, 
      Table[λ[k], {k, 1, 2}]]

Solve[(∑_(k = 2)^3) λ[k]*v[k] == 0, 
      Table[λ[k], {k, 2, 3}]]

Solve[(∑_(k = 3)^4) λ[k]*v[k] == 0, 
      Table[λ[k], {k, 3, 4}]]

Dim(Bild(f))=2

Aufgabe 63

A = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}

Transpose[A]

Aufgabe 64

RechtsSumme[f_, {x_, a_, b_}, 
        n_] := 
(∑_(k = 1)^n)(f /. 
              x -> a + ((k (b - a))/n)) ((b - a)/n)

s = RechtsSumme[(x^2), {x, 0, 1}, n]

Limit[s, n -> Infinity]

Aufgabe 65

Hospital[f_, g_, x_ -> x0_] := 
      If[Limit[f, x -> x0] == 0  &&  
          Limit[g, x -> x0] == 0, 
        Limit[D[f, x]/D[g, x], 
          x -> x0], 
Limit[f, x -> x0]/
          Limit[g, x -> x0]]

Hospital[Log[1 + (x/n)], (1/n), n -> ∞]

Aufgabe 66

Plot[Sin[x], {x, 0, 2  π}]

(∫_0^(2  π))Abs[Sin[x]] \[DifferentialD]x

Plot[Abs[x], {x, -1, 2}]

(∫_(-1)^2) Abs[x] \[DifferentialD]x

Plot[Log[x], {x, 1, 2}]

(∫_1^2) Log[x] \[DifferentialD]x

Plot[(x^5), {x, 0, 1}, PlotRange -> All]

(∫_0^1) (x^5) \[DifferentialD]x