# 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.

Question
Matrix transformations
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.

2021-01-26
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.$$

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