next up previous contents
Next: Change of basis. Up: No Title Previous: Matrix inversion.

More on systems of linear equations.

Conisder the following system of equations (as in  1, with n=p):

 \begin{displaymath}
\begin{cases}
a_{11}x_1+ a_{12}x_2+ a_{13}x_3+ \dots + a_{1n...
...}x_1+ a_{n2}x_2+ a_{n3}x_3+ \dots + a_{nn}x_n =b_n
\end{cases}\end{displaymath} (8.1)

To this system we associate the restricted matrix (v.s.  3) of the system:


\begin{displaymath}A=
\begin{pmatrix}a_{11} & a_{12} &a_{13} & \dots & a_{1n} \\...
...s \\
a_{n1} & a_{n2} &a_{n3} & \dots & a_{nn}
\end{pmatrix}\end{displaymath}

The system  8 can be written in matricial form:


\begin{displaymath}\begin{pmatrix}a_{11} & a_{12} &a_{13} & \dots & a_{1n} \\
...
...atrix}=
\begin{pmatrix}b_1 \\ b_2 \\ \dots \\ b_n \end{pmatrix}\end{displaymath}

or in a shorter way: AX=B, where A is the (restricted) matrix of the system, $X=\begin{pmatrix}x_1 \\ x_2 \\ \dots \\ x_n \end{pmatrix}$ and $B=\begin{pmatrix}b_1 \\ b_2 \\ \dots \\ b_n \end{pmatrix}$.

How to solve some square linear systems?

\fbox{
$ AX=B \Longleftrightarrow X= A^{-1}B $ }



Noah Dana-Picard
2001-02-26