Show that W, the set of all 3 times 3 upper triangular matrices, forms a subspace of all 3 times 3 matrices. What is the dimension of W? Find a basis for W.

To show that W, the set of all $3\times 3$ upper triangular matrices,
forms asubspace of all $3\times 3$ matrices.
Now we know that M, the set of all $3\times 3$ matrices, forms a vector space.
Let A, B in W with $A=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ 0& d_1& c_1\\ 0& 0& f_1\end{array}\right]B=\left[\begin{array}{ccc}{a}_2& b_2& c_2\\ 0& d_2& c_2\\ 0& 0& f_2\end{array}\right]$
and k be a scalar. Now,
$A+B=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ 0& d_1& c_1\\ 0& 0& f_1\end{array}\right]+\left[\begin{array}{ccc}{a}_2& b_2& c_2\\ 0& d_2& c_2\\ 0& 0& f_2\end{array}\right]=\left[\begin{array}{ccc}{a}_1+a_2& b_1+b_2& c_1+c_2\\ 0& d_1+d_2& c_1+c_2\\ 0& 0& f_1+f_2\end{array}\right]$
Once again, $kA=k\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ 0& d_1& c_1\\ 0& 0& f_1\end{array}\right]$
$\left[\begin{array}{ccc}k{a}_1& k{b}_1& k{c}_1\\ 0& k{d}_1& k{c}_1\\ 0& 0& k{f}_1\end{array}\right]$ in W
Therefore, W is a subspace of M. Now let x is a typical element of W, with
$X=\left[\begin{array}{c}abc\\ 0dc\\ 00f\end{array}\right]$
can be written as $A=a{E}^{11}+b{E}_{12}+c{E}_{13}+d{E}_{22}+c{E}_{23}+f{E}_{33}$
where $E_{ij}$ is
$3\times 3$ time matrix with (i, j) element is 1 and rest are zore.
Now for any linear combination
$x_{1}E11+x_{2}E12+x_{3}E13+x_{4}E22+x_{5}E23+x_{6}E33=0_{3\times 3}$
imply $x_1=x_2=\cdots =x_6=0$
Therefore, $E^{11},E_{12},E_{13},E_{22},E_{23},E_{33}$ forms a basis of W,
hence dim $W=6.$