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.

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

To show that W, the set of all $3×3$ upper triangular matrices,
forms asubspace of all $3×3$ matrices.
Now we know that M, the set of all $3×3$ 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
$3×3$ time matrix with (i, j) element is 1 and rest are zore.
Now for any linear combination
${x}_{1}E11+{x}_{2}E12+{x}_{3}E13+{x}_{4}E22+{x}_{5}E23+{x}_{6}E33={0}_{3×3}$
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 $W=6.$