91Ó°ÊÓ

All diagonal matrices in \(\mathrm{M}_{2 \times 2}(F)\) have the form: \begin{equation} A = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \end{equation} where \(a, b \in F\).

We want to find a set of diagonal matrices that are linearly independent and can generate all other diagonal matrices through linear combinations. Let's consider the following two diagonal matrices: \[ A_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad A_2 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \] Now let's see if they are linearly independent and can generate all other diagonal matrices: 1. Linearly Independent: There is no scalar \(c_1, c_2 \in F\) such that \(c_1A_1 + c_2A_2 = 0\) (the zero matrix) unless both \(c_1\) and \(c_2\) are zero. Proof: If there exist \(c_1, c_2 \in F\) such that \(c_1A_1 + c_2A_2 = 0\), then: \[ c_1 \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + c_2 \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \] This implies \(c_1 = 0\) and \(c_2 = 0\), so the matrices \(A_1\) and \(A_2\) are linearly independent. 2. Generates Subspace: For any diagonal matrix \(A = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\), we can find scalars \(c_1, c_2 \in F\) such that \(A = c_1A_1 + c_2A_2\) as follows: \[ c_1 \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + c_2 \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \] Setting \(c_1 = a\) and \(c_2 = b\) accomplishes this: \[ a \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + b \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \] Therefore, the set \(\left\{ A_1, A_2 \right\}\) is a linearly independent set that generates the subspace of diagonal matrices in \(\mathrm{M}_{2 \times 2}(F)\).