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 \(A=\begin{bmatrix}a_{1} & b_{1} & c_{1} \\0 & d_{1} & c_{1}\\ 0 & 0 & f_{1} \end{bmatrix} B=\begin{bmatrix}a_{2} & b_{2} & c_{2} \\0 & d_{2} & c_{2}\\ 0 & 0 & f_{2} \end{bmatrix}\)

and k be a scalar. Now,

\(A+B=\begin{bmatrix}a_{1} & b_{1} & c_{1} \\0 & d_{1} & c_{1}\\ 0 & 0 & f_{1} \end{bmatrix}+\begin{bmatrix}a_{2} & b_{2} & c_{2} \\0 & d_{2} & c_{2}\\ 0 & 0 & f_{2} \end{bmatrix}=\begin{bmatrix}a_{1} + a_{2} & b_{1} +b_{2} & c_{1} + c_{2} \\0 & d_{1}+d_{2} & c_{1}+c_{2}\\ 0 & 0 & f_{1}+f_{2} \end{bmatrix}\)

Once again, \(kA=k\begin{bmatrix}a_{1} & b_{1} & c_{1} \\0 & d_{1} & c_{1}\\ 0 & 0 & f_{1} \end{bmatrix}\)

\(\begin{bmatrix}ka_{1} & kb_{1} & kc_{1} \\0 & kd_{1} & kc_{1}\\ 0 & 0 & kf_{1} \end{bmatrix}\) in W

Therefore, W is a subspace of M. Now let x is a typical element of W, with

\(X=\begin{bmatrix}abc \\0dc\\ 00f \end{bmatrix}\)

can be written as \(A = aE^{11} + bE_{12} + cE_{13} + dE_{22} + cE_{23} + fE_{33}\)

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 \(W = 6.\)

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 \(A=\begin{bmatrix}a_{1} & b_{1} & c_{1} \\0 & d_{1} & c_{1}\\ 0 & 0 & f_{1} \end{bmatrix} B=\begin{bmatrix}a_{2} & b_{2} & c_{2} \\0 & d_{2} & c_{2}\\ 0 & 0 & f_{2} \end{bmatrix}\)

and k be a scalar. Now,

\(A+B=\begin{bmatrix}a_{1} & b_{1} & c_{1} \\0 & d_{1} & c_{1}\\ 0 & 0 & f_{1} \end{bmatrix}+\begin{bmatrix}a_{2} & b_{2} & c_{2} \\0 & d_{2} & c_{2}\\ 0 & 0 & f_{2} \end{bmatrix}=\begin{bmatrix}a_{1} + a_{2} & b_{1} +b_{2} & c_{1} + c_{2} \\0 & d_{1}+d_{2} & c_{1}+c_{2}\\ 0 & 0 & f_{1}+f_{2} \end{bmatrix}\)

Once again, \(kA=k\begin{bmatrix}a_{1} & b_{1} & c_{1} \\0 & d_{1} & c_{1}\\ 0 & 0 & f_{1} \end{bmatrix}\)

\(\begin{bmatrix}ka_{1} & kb_{1} & kc_{1} \\0 & kd_{1} & kc_{1}\\ 0 & 0 & kf_{1} \end{bmatrix}\) in W

Therefore, W is a subspace of M. Now let x is a typical element of W, with

\(X=\begin{bmatrix}abc \\0dc\\ 00f \end{bmatrix}\)

can be written as \(A = aE^{11} + bE_{12} + cE_{13} + dE_{22} + cE_{23} + fE_{33}\)

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 \(W = 6.\)