Editing Triangular matrix (section)

===Forward substitution===
The matrix equation ''L'''''x''' = '''b''' can be written as a system of linear equations

:<math>\begin{matrix}
  \ell_{1,1} x_1 &   &                &   &        &   &                & = &    b_1 \\
  \ell_{2,1} x_1 & + & \ell_{2,2} x_2 &   &        &   &                & = &    b_2 \\
          \vdots &   &         \vdots &   & \ddots &   &                &   & \vdots \\
  \ell_{m,1} x_1 & + & \ell_{m,2} x_2 & + & \dotsb & + & \ell_{m,m} x_m & = &    b_m \\
\end{matrix}</math>

Observe that the first equation (<math>\ell_{1,1} x_1 = b_1</math>) only involves <math>x_1</math>, and thus one can solve for <math>x_1</math> directly. The second equation only involves <math>x_1</math> and <math>x_2</math>, and thus can be solved once one substitutes in the already solved value for <math>x_1</math>. Continuing in this way, the <math>k</math>-th equation only involves <math>x_1,\dots,x_k</math>, and one can solve for <math>x_k</math> using the previously solved values for <math>x_1,\dots,x_{k-1}</math>. The resulting formulas are:
:<math>\begin{align}
  x_1 &= \frac{b_1}{\ell_{1,1}}, \\
  x_2 &= \frac{b_2 - \ell_{2,1} x_1}{\ell_{2,2}}, \\
      &\ \ \vdots \\
  x_m &= \frac{b_m - \sum_{i=1}^{m-1} \ell_{m,i}x_i}{\ell_{m,m}}.
\end{align}</math>

A matrix equation with an upper triangular matrix ''U'' can be solved in an analogous way, only working backwards.