Mengenzugehörigkeit, Probleme bei Gleichungssystemen mit Parametern, Aussonderung reeller n-ter Wurzeln aus den komplexen Wurzeln u.s.w.

Probiere aus:

1 Umgang mit Mengen und Einschränkung auf Mengen

?Assuming

?Refine

Refine[Abs[a],a≤0]

Refine[((((b^r)))^p), 
      b ∈ Reals && (r/2) ∈ Integers]

Refine[Cos[x+πy],y∈Integers]

s^3+6 s^2 Sqrt[s^2-(2 s^2/4)]//Simplify

Assuming[s > 0 && s ∈ Reals, 
      s^3 + 6 s^2 Sqrt[s^2 - (2 s^2/4)] // Simplify]

2 Problem des Lösungsverhaltens abhängig von Parametern

Remove["Global`*"]

A = {{1, 2, -1}, {2, -6, 6 - x}, {x, -4, 
            2}};
Det[A] // Factor

A // MatrixForm

NullSpace[A]

?NullSpace

B = {{1, 2, -1}, {2, -6, 6 - 3}, {3, -4, 
            2}};

b = NullSpace[B]

Transpose[b]

B . Transpose[b]

A . {x1, x2, x3} == {1, 2, 3}

solv = 
      Solve[A . {x1, x2, x3} == {1, 2, 3}, {x1, x2, x3}] // 
        Flatten

vM[t_, x_] := {{x1}, {x2}, {x3}} + t Transpose[NullSpace[B]] /. 
        solv

vM[t, x] /. x -> 3

(A /. x -> 3)

Det[(A /. x -> 3)]

(A /. x -> 3) . {x1, x2, x3} == {1, 2, 3}

solv = 
      Solve[(A /. x -> 3) . {x1, x2, x3} == {1, 2, 3}, {x1, 
            x2, x3}] // Flatten

{x1, x2, x3} /. solv

Remove[x1, x2, x3]

Reduce[A . {x1, x2, x3} == {1, 2, 3}]

x

RowReduce[A]

RowReduce[A /. x -> 3]

LinearSolve[A, {1, 2, 3}]

Append[Transpose[A], {1, 2, 3}] // Transpose // 
      MatrixForm

Append[Transpose[A], {1, 2, 3}] // Transpose // RowReduce // 
      MatrixForm

solv = 
      LinearSolve[
        Evaluate[Refine[A, Det[A] == 0]], {1, 2, 3}]

solv /. x -> 3

Refine[A, Det[A] == 0]

Assuming[A, Det[A] == 0]

Assuming[A, Det[A] =!= 0]

LinearSolve[Assuming[A, Det[A] =!= 0], {1, 2, 3}]

x/x

Solve[A . {x1, x2, x3} == {1, 2, 3}, {x1, x2, x3}, 
      VerifySolutions -> False]

?=!=

3

Remove["Global`*"]

A[x_]:={{x+1,2,-1},{2,0,2},{0,x+1,-1}}

A[x]//MatrixForm

Det[A[x]]//Factor

Solve[Det[A[x]]==0,{x}]

Solve[A[x].{a,b,c}=={u,v,w},{a,b,c}]

Solve[A[x].{a,b,c}=={1,2,3},{a,b,c}]

Remove["Global`*"]

A[x_]:={{1,2,-1},{2,-6,6-x},{x,-4,2}}

A[x]//MatrixForm

Det[A[x]]//Factor

Solve[Det[A[x]]==0,{x}]

Solve[A[x].{a,b,c}=={u,v,w},{a,b,c}]

Solve[A[x].{a,b,c}=={u,v,w},{a,b,c}]/.x->3

Solve[A[2].{a,b,c}=={u,v,w},{a,b,c}]

Solve[A[3].{a,b,c}=={u,v,w},{a,b,c}]

Solve[A[x].{a,b,c}=={1,2,3},{a,b,c}]

Solve[A[x].{a,b,c}=={1,2,4},{a,b,c}]

?NullSpace

Assuming[Det[A[x]]==0,NullSpace[A[x]]]

Assuming[x==1,NullSpace[A[x]]]

?LinearSolve

LinearSolve[A[x],{1,2,3}]

Assuming[Det[A[x]]==0,LinearSolve[A[x],{1,2,3}]]

Solve[A[x].{a,b,c}=={1,2,3},{a,b,c}]//Simplify

Assuming[Det[A[x]]==0,Solve[A[x].{a,b,c}=={1,2,3},{a,b,c}]//Simplify]

4 Umgang mit komplexen Wurzeln, Aussonderung der reellen Wurzeln u.s.w.

Remove["Global`*"]

Factor[2-x^2]

Factor[2-x^2,Extension->Sqrt[2]]

?Root

rr = Root[24 - 2x+ 4x^4, 1]

ToRadicals[%]

N[%]

rr = Root[24 - 2x+ 4x^4, 2]//ToRadicals

N[%]

Table[ToRadicals[Root[24 - 2x+ 4x^4, k]],{k,1,4}]

Table[ToRadicals[Root[24 - 2x+ 4x^4, k]],{k,1,4}]//N

Table[ToRadicals[Root[24 - 4x^4, k]],{k,1,4}]

Table[
      If[
          Element[ToRadicals[Root[24 - 4x^4, k]],Reals],
          b[k]=ToRadicals[Root[24 - 4x^4, k]], b[k]=.
        ]
     ,{k,1,4}]