In a linear algebra problem we are asked to prove that pre-multiplying a matrix \(\displaystyle A_m\) by the elementary matrix obtained with any matrix elementary line transformation (ex: \(\displaystyle I_m \underset{l_1 \leftrightarrow l_2} \longrightarrow E \)) is the same as applying said elementary line transformation on the matrix \(\displaystyle A_m\).
This is fairly simple to illustrate:
\(\displaystyle
E=
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}
= I_m \underset{l_1 \leftrightarrow l_2} \longrightarrow E;
\hspace{10px}
A=
\begin{pmatrix}
1 & 2 \\
2 & 3 \\
\end{pmatrix}
\)
\(\displaystyle
E.A =
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}
.
\begin{pmatrix}
1 & 2 \\
2 & 3 \\
\end{pmatrix}
=
\begin{pmatrix}
2 & 3 \\
1 & 2 \\
\end{pmatrix}
=
A \underset{l_1 \leftrightarrow l_2} \longrightarrow EA;
\)
While I can clearly understand the concepts and ilustrate them, I tend to have a real hard time actually writing the "formal proof" for cases like this.
Can some one point me in the right direction both in this case and when approaching similar cases?
Cheers,
Diogo
P.S: Between searching LaTeX formats and writing them, sometimes I spend 20+ minutes elaborating an O.P...Is it normal, or am I doing something wrong?
This is fairly simple to illustrate:
\(\displaystyle
E=
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}
= I_m \underset{l_1 \leftrightarrow l_2} \longrightarrow E;
\hspace{10px}
A=
\begin{pmatrix}
1 & 2 \\
2 & 3 \\
\end{pmatrix}
\)
\(\displaystyle
E.A =
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}
.
\begin{pmatrix}
1 & 2 \\
2 & 3 \\
\end{pmatrix}
=
\begin{pmatrix}
2 & 3 \\
1 & 2 \\
\end{pmatrix}
=
A \underset{l_1 \leftrightarrow l_2} \longrightarrow EA;
\)
While I can clearly understand the concepts and ilustrate them, I tend to have a real hard time actually writing the "formal proof" for cases like this.
Can some one point me in the right direction both in this case and when approaching similar cases?
Cheers,
Diogo
P.S: Between searching LaTeX formats and writing them, sometimes I spend 20+ minutes elaborating an O.P...Is it normal, or am I doing something wrong?
