# Find a basis for the space of 2 \times 2

Find a basis for the space of $2×2$ lower triangular matrices
Basis =
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Thomas Conway

S = set of lower triangular matrices.
To find basis of s
Let $A=\left[\begin{array}{cc}a& 0\\ b& c\end{array}\right]ϵS$
$A=a\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+b\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]+c\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
$\therefore$ A can be written as the linear combination of vectors
$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]and\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
and $B=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
is linearly independent set
$\therefore$ B is basis for the space of $2×2$ lower triangular matrices
$B=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$

Jeffrey Jordon