Let W be the subspace of all diagonal matrices in M2,2. Find a bais for...

Step 1
Given that W is the subspace of all diagonal matrices in $M_{2,2}$
The objective is to a basis for W.
Step 2
Consider the given vector space W of all diagonal matrices in $M_{2,2}$
Therefore,
$W=\{\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right],\text{a and b can be any real number}\}$
Let E be basis for W.
Therefore,
$\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]=a\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+b\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$
Now, let ${\lambda }_1$ and ${\lambda }_2$ be any two scalars such that:
${\lambda }_1\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+{\lambda }_2\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]=[]$
$\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& {\lambda }_2\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\left[\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
Equating the elements:
${\lambda }_1=0,{\lambda }_2=0$
Therefore, $\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$ and $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$ are linear independent.
Hence, the basis of W is $E=\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\}$ and dimension is 2.

Answer is given below (on video)