91Ó°ÊÓ

The diagonal matrices in the vector space of 2x2 matrices with entries in the field F have the structure: \[\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\] where a, b are elements of the field F.

We can see that only the diagonal elements can be any non-zero value and be a diagonal matrix. We can then define two matrices, \(D_1\) and \(D_2\), as our generating set elements: \[D_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\] \[D_2 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\]

Let's show that the set \(\{D_1, D_2\}\) is linearly independent. To do this, we can consider the linear combination of the set elements, \(D_1\) and \(D_2\), and set the linear combination equal to the zero matrix to form a homogeneous system: \[c_1D_1 + c_2D_2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\] Substitute the values of \(D_1\) and \(D_2\) to get: \[c_1\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + c_2\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\] This implies: \[c_1(1) + c_2(0) = 0\] and \[c_1(0) + c_2(1) = 0\] We can easily see that the only solution to this system is the trivial solution of \(c_1 = c_2 = 0\). Thus the set \(\{D_1, D_2\}\) is linearly independent.

To show that \(\{D_1, D_2\}\) generates the diagonal matrix subspace of M_{2x2}(F), we can simply use linear combinations of \(D_1\) and \(D_2\) to create any other diagonal matrix in M_{2x2}(F). Let \(D = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\) be any diagonal matrix in M_{2x2}(F), then the linear combination: \[aD_1 + bD_2 = a\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + b\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} = D\] - shows that \(\{D_1, D_2\}\) generates the diagonal matrix subspace. Thus, the linearly independent set that generates the subspace of diagonal matrices in \(\mathrm{M}_{2 \times 2}(F)\) is: \[\{D_1, D_2\} = \left\{\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\right\}.\]