Eine positive quadratische Form: Paraboloid mit Minimum

Plot3D[(x^2) + (y^2), {x, -1, 1}, {y, -1, 1}]

Ein indefinite quadratische Form: Hyperbolisches Paraboloid mit Sattelpunkt

Plot3D[(x^2) - (y^2), {x, -1, 1}, {y, -1, 1}]

Plot3D[x y, {x, -1, 1}, {y, -1, 1}]

Gram-Schmidtsches Orthogonalisierungsverfahren

Länge[v_] := 
      Sqrt[(∑_(k = 1)^(Length[v]))v[[k]]^2]

Clear[GramSchmidt]
GramSchmidt[A__] := 
      Module[{v, w, k, 
          m}, 
v = {A}; 
Do[
w[m] = 
            v[[m]] - (∑_(k = 1)^(m - 1))(v[[m]] . w[k])*
                  w[k]; 
w[m] = 
            w[m]/Länge[w[m]], {m, 1, 
            Length[v]}]; 
Table[
          w[k],  {k, 1, Length[v]}]
]

GramSchmidt[{3, 4}, {1, 0}]

GramSchmidt[{1, 0, 0}, {1, 1, 0}, {1, 1, 1}]

GramSchmidt[{1, 1, 1}, {1, 1, 0}, {1, 0, 0}]

GramSchmidt[{1, 2, 3, 4, 5}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 
        1, 0, 0}, {0, 0, 0, 1, 0}]

Clear[GramSchmidtOhneNormierung]
GramSchmidtOhneNormierung[A__] := 
      Module[{v, w, k, 
          m}, 
v = {A}; 
Do[
w[m] = 
            v[[m]] - (∑_(k = 1)^(m - 1))v[[m]] . w[k]/Länge[w[k]]^2*w[k], 
{m, 1, 
            Length[v]}]; 
Table[
          w[k],  {k, 1, Length[v]}]
]

GramSchmidtOhneNormierung[{1, 0, 0}, {1, 1, 0}, {1, 1, 1}]

GramSchmidtOhneNormierung[{1, 1, 1}, {1, 1, 0}, {1, 0, 0}]

GramSchmidtOhneNormierung[{1, 2, 3, 4, 5}, {1, 0, 0, 0, 0}, {0, 1, 0, 
        0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}]

Legendre-Polynome

Table[LegendreP[n, t], {n, 0, 10}]

Skalarprodukt[f_, g_, 
        t_] := (∫_(-1)^1) f*g \[DifferentialD]t

Table[
      Skalarprodukt[LegendreP[i, t], LegendreP[j, t], t], {i, 0, 10}, {j, 0, 
        10}]

Gramsche Determinante

GramMatrix[A_] := 
      Table[A[[i]] . A[[j]], {i, Length[A]}, {j, 
          Length[A]}]
GramDeterminant[A_] := Det[GramMatrix[A]]

GramMatrix[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}]

GramMatrix[{{1, 1, 1}, {1, 1, 0}, {1, 0, 0}}]

GramDeterminant[{{1, 1, 1}, {1, 1, 0}, {1, 0, 0}}]