Suman Cole

2021-01-28

Let W be the subspace of all diagonal matrices in ${M}_{2,2}$. Find a bais for W. Then give the dimension of W.
If you need to enter a matrix as part of your answer , write each row as a vector.For example , write the matrix

Tasneem Almond

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,
$W=\left\{\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right],\text{a and b can be any real number}\right\}$
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:
${\lambda }_{1}\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+{\lambda }_{2}\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]=\left[\right]$
$\left[\begin{array}{cc}{\lambda }_{1}& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 0& {\lambda }_{2}\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
$\left[\begin{array}{cc}{\lambda }_{1}& 0\\ 0& {\lambda }_{2}\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
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 $E=\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}$ and dimension is 2.

Jeffrey Jordon