Homogenous Systems of LE

Definition

A system of linear equations is homogeneous when all constant terms are zero. It follows this general form:

$$\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=0\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=0\\ ⋮\\ a_{m1}{x}_1+a_{m2}{x}_2+\cdots +a_{mn}{x}_n=0\end{array}$$

A homogeneous system always has at least one solution—the trivial solution, where all variables equal zero.

Example: Solving a Homogeneous System

Problem: Solve the system:

$$\{\begin{array}{l}{x}_1-x_2+3x_3=0\\ 2x_1+x_2+3x_3=0\end{array}$$

Solution Process: Using Gauss-Jordan elimination:

1. Convert the system into a matrix:

   $$\left[\begin{array}{cccc}1& -1& 3& 0\\ 2& 1& 3& 0\end{array}\right]$$

2. Perform row operations to simplify:

   • $R_2\leftarrow R_2-2R_1$
   • $R_2\leftarrow \frac{1}{3}{R}_2$
   • $R_1\leftarrow R_1+R_2$

   Resulting matrix:

   $$\left[\begin{array}{cccc}1& 0& 2& 0\\ 0& 1& -1& 0\end{array}\right]$$

3. Convert back to equations:

   $$\{\begin{array}{l}{x}_1+2x_3=0\\ x_2-x_3=0\end{array}$$

4. Express solution using a parameter:

   • Let $x_3=t$
   • Then: $x_1=-2t$, $x_2=t$, $x_3=t$
   • The system has infinitely many solutions, including the trivial solution where $t=0$.

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linear math