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A special case: homogeneous equations.

When all the entries in the last column of the augmented matrix are equal to 0, the system is called an homogeneous system. Such a sytem has always at least one solution, called the trivial solution, i.e. $(0,0, \dots , 0)$.

The question which remains to be solved is: `` are there other solutions?''. First we will see examples.

Remark 3.3.6   For an homogeneous system, we do not need to use the augmented matrix. As the last colum of the augmented matrix is made only of zeros, any elementary row operation will leave it unchanged. Thus row reducing of the restricted matrix will be enough.

Example 3.3.7   We solve the system: $\begin{cases}x -2y+3z =0 \\ 2x -y +z =0 \\ x + 5y -4z =0 \end{cases}$.

The restricted matrix of this homogeneous system is $A=\begin{matrix}1 & -2 & 3 \\ 2 & -1 & 1 \\ 1 & 5 & -4 \end{matrix}$. We row reduce it:
\begin{align*}\begin{pmatrix}1 & -2 & 3 \\ 2 & -1 & 1 \\ 1 & 5 & -4 \end{pmatrix...
...R_2}\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{align*}
This last matrix is the restricted matrix of an homogeneous system which has only the trivial solution.

Example 3.3.8   We solve the system: $\begin{cases}x -y+3z -2t =0 \\ 2x -3y +z+3t =0 \\ x + 5y -4z+t =0 \end{cases}$.

The restricted matrix of this homogeneous system is $A=\begin{pmatrix}1 & -1 & 3 &-2 \\ 2 & -3 & 1 & 3 \\ 1 & 5 & -4 & 1 \end{pmatrix}$.

Let us row-reduce the matrix A:
\begin{align*}\begin{pmatrix}1 & -1 & 3 &-2 \\ 2 & -3 & 1 & 3 \\ 1 & 5 & -4 & 1 ...
...& 0 & -\frac {34}{37} \\ 0 & 0 & 1 & -\frac {45}{37} \end{pmatrix}
\end{align*}
The colums corresponding to the unknowns x, y and z contains pivots, thus these unknowns are principal. The last unknown (i.e. t) is free; we obtain the principal unknowns as functions of t:

\begin{displaymath}\begin{cases}
x =\frac {27}{37}t \\ y=\frac {34}{37}t \\ z=\frac {45}{37}t
\end{cases}\end{displaymath}

The set of solutions of the given system is thus:

\begin{displaymath}\mathcal{S}= \left\{ \left( \frac {27}{37}t, \frac {34}{37}t, \frac {45}{37}t , t
\right) \; ; \; t \in \mathbb{R}\right\}
\end{displaymath}

We can make a remark, which will be useful for the next chapter: define t=37u, for $u
\in \mathbb{R} $. Then we have:

\begin{displaymath}\mathcal{S}=\{ ( 27u, 34u,45u,37u )\; ; \; u \in \mathbb{R}\}
\end{displaymath}


next up previous contents
Next: The main theorems. Up: Solution of a system Previous: The general case.
Noah Dana-Picard
2001-02-26