Linear Equations System Solving Task

A Linear Equations System Solving Task is a system of equations solving task that accepts a linear equation system.

• AKA: Solving a System of Linear Equations.
• Context:
  • It can be solved by a Linear Equation System Solver (that implements a linear equation system solving algorithm).
  • It can be solved by graphing all the equations and finding the common points of all the graphs.
  • It can be solved by Gauss Elimination method.
  • It can be solved by Matrix Inversion method.
• Example(s):
  • for linear equation system [math]\displaystyle{ \begin{array}{lcl} 2x_1-x_2 & = & 1 \\ x_1+x_2 & = &2 \end{array} }[/math] Geometrically the solution is about finding a point(s) which is(are) intersection of the two given straight lines [math]\displaystyle{ 2x_1-x_2=1 }[/math] and [math]\displaystyle{ x_1+x_2=2 }[/math]. The above system of linear equation in matrix form can be written as[math]\displaystyle{ \begin{bmatrix}2& -1\\1& 1\end{bmatrix}\begin{bmatrix} x_1\\x_2 \end{bmatrix}=\begin{bmatrix} 1\\2 \end{bmatrix} }[/math]

  • for linear equation system [math]\displaystyle{ \begin{array}{lcl} x_1+x_2+x_3 & = & 3 \\ x_1-x_2-x_3 & = & 1 \\ x_1+2x_2+3x_3 & = & 4 \end{array} }[/math] Geometrically the solution is about finding a point(s) which is(are) intersection of the three given planes [math]\displaystyle{ x_1+x_2+x_3=3 }[/math], [math]\displaystyle{ x_1-x_2-x_3=1 }[/math] and [math]\displaystyle{ x_1+2x_2+3x_3=4 }[/math]. The above system of linear equation in matrix form can be written as[math]\displaystyle{ \begin{bmatrix}1& 1& 1\\1& -1& -1\\1& 2& 3\end{bmatrix}\begin{bmatrix} x_1\\x_2\\x_3 \end{bmatrix}=\begin{bmatrix} 3\\1\\4 \end{bmatrix} }[/math]

• Counter-Example(s):
  • a Non-Linear Equation Solving Task, such as quadratic equation system solving.
  • a Linear Programming Task.
• See: Linear Algebra.

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References

2007

• http://www.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html
  • QUOTE: Mathematical Challenge Eight - Beyond Convex Optimization: Can linear algebra be replaced by algebraic geometry in a systematic way?

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