Again let \(V\) be the space of \(2 \times 2\) matrices over \(F\). Find a basis \(\{A_1, A_2, A_3, A_4\}\) for \(V\) such that \(A_j^2 = A_j\) for each \(j\).
Solution
1. Define four candidate matrices in \(M_2(F)\):\[ A_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad A_2 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}, \quad A_3 = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}, \quad A_4 = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}. \]
explicit definitions.
2. Verify that each matrix is idempotent (\(A_j^2 = A_j\)):
2.1. \(A_1^2 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = A_1\).
matrix multiplication.
2.2. \(A_2^2 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = A_2\).
matrix multiplication.
2.3. \(A_3^2 = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} = A_3\).
matrix multiplication.
2.4. \(A_4^2 = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} = A_4\).
matrix multiplication.
3. Prove linear independence of \(\{A_1, A_2, A_3, A_4\}\):
3.1. Suppose \(c_1 A_1 + c_2 A_2 + c_3 A_3 + c_4 A_4 = 0\) for scalars \(c_1, c_2, c_3, c_4 \in F\).
setup.
3.2. Writing this linear combination as a single matrix yields:\[ \begin{pmatrix} c_1 + c_3 + c_4 & c_3 \\ c_4 & c_2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}. \]
entrywise matrix addition and scalar multiplication.
3.3. Equating corresponding entries:- Entry \((1, 2) \implies c_3 = 0\).- Entry \((2, 1) \implies c_4 = 0\).- Entry \((2, 2) \implies c_2 = 0\).- Entry \((1, 1) \implies c_1 + c_3 + c_4 = 0 \implies c_1 = 0\).
linear system of equations.
3.4. Since \(c_1 = c_2 = c_3 = c_4 = 0\), \(\{A_1, A_2, A_3, A_4\}\) is linearly independent.
definition of linear independence.
4. Conclusion:Since \(\dim(M_2(F)) = 4\), any linearly independent set of 4 matrices forms a basis.Thus \(\{A_1, A_2, A_3, A_4\}\) is a basis of idempotent matrices for \(M_2(F)\). Q.E.D.
2 and 3.