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11TH•Mathematics•Matrices And Determinants•4.9 Homogeneous Systems

4.9 System of Homogeneous Linear Equations

Homogeneous linear systems, trivial and non-trivial solutions, Gaussian elimination.

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4.9 System of Homogeneous Linear Equations

The system

a11x1+a12x2+a13x3=0a21x1+a22x2+a23x3=0a31x1+a32x2+a33x3=0\begin{align*} a_{11}x_1 + a_{12}x_2 + a_{13}x_3 &= 0 \\ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 &= 0 \\ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 &= 0 \end{align*}

always has the solution (0,0,0)(0,0,0), so it is always consistent.

  • Trivial solution: (0,0,0)(0,0,0)
  • Non-trivial solution: any other solution

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