
Originally Posted by
Craka
The vectors are (1,-2,3) (-1,3,2) and (-1,10,5) . Question asks to find if they are linearly independent.
So I've put them into matrix form, as below
$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
{ - 2} & 3 & {10} \\
3 & 2 & 5 \\
\end{array}} \right]
$
now start gauss-jordon elimination.
new row2 = old row2 + 2*row1
$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
3 & 2 & 5 \\
\end{array}} \right]
$
new row3 = old row3 - 3*row1
$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
0 & 5 & 8 \\
\end{array}} \right]
$
new row3 = old row3 - 5&row2
$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
0 & 0 & { - 32} \\
\end{array}} \right]
$
new row3 = old row3 divided by 32
$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
0 & 0 & 1 \\
\end{array}} \right]
$
new row2 = old row2 - 8*row3
$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]
$
new row1 = old row1 + row3
$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]
$
new row1 = old row1 + row 2
$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]
$
Firstly is my working correct, and secondly I would think they a linearly independent for the fact that there is only one 1 in each column am I correct in thinking this?