MATH347 L7: Systems of linear equations

• New concepts:

  • Bordered matrix for a linear system

  • Similarity transformations of a linear system

  • Gaussian elimination

Gaussian elimination: Reduction to triangular form, backsubstitution

• Idea: make one fewer unknown appear in each equation. Use first equation to eliminate $x_1$ in equations 2,3

  ${\displaystyle \{\begin{array}{ccc}{x}_1+2x_2-x_3& =& 2\\ 2x_1-x_2+x_3& =& 2\\ 3x_1-x_2-x_3& =& 1\end{array}.\Rightarrow \{\begin{array}{ccc}{x}_1+2x_2-x_3& =& 2\\ -5x_2+3x_3& =& -2\\ -7x_2+2x_3& =& -5\end{array}.}$

• Use second equation to eliminate $x_2$ in equation 3

  ${\displaystyle \{\begin{array}{ccc}{x}_1+2x_2-x_3& =& 2\\ -5x_2+3x_3& =& -2\\ -7x_2+2x_3& =& -5\end{array}.\Rightarrow \{\begin{array}{ccc}{x}_1+2x_2-x_3& =& 2\\ -5x_2+3x_3& =& -2\\ -\frac{11}{5}{x}_3& =& -\frac{11}{5}\end{array}.}$

• Start finding components from last to first to obtain $x_3=1$, $x_2=1$,$x_1=1$

Matrix similarity transformations: reduce to triangular form

• Explicitly writing the unknowns $x_1,x_2,x_3$ is not necessary. Intoduce the “bordered” matrix

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

• Define allowed operations:

  • multiply a row by a non-zero scalar

  • add a row to another

• Bordered matrices obtained by the allowed operations are said to be similar, in that the solution of the linear system stays the same

  ${\displaystyle \left[\begin{array}{cccc}1& 2& -1& 2\\ 2& -1& 1& 2\\ 3& -1& -1& 1\end{array}\right]\sim \left[\begin{array}{cccc}1& 2& -1& 2\\ 0& -5& 3& -2\\ 0& -7& 2& -5\end{array}\right]\sim \left[\begin{array}{cccc}1& 2& -1& 2\\ 0& -5& 3& -2\\ 0& 0& -\frac{11}{5}& -\frac{11}{5}\end{array}\right]}$

Two linear systems: same system matrix, different right hand sides

Form the bordered matrix in both cases, and reduce to triangular form