Worksheet 26

1. Prove the complex conjugation is a homomorphism with respect to multiplication of complex numbers. Precisely, show that for any $w,z \in \mathbb{C}$ we have \(\overline{wz} = (\overline{w})(\overline{z})\) In other words, the product of the complex conjugates is equal to the complex conjugate of the product.

2. Determine whether the matrix \(A = \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}\) has a basis of eigenvectors over the fields:
   • $\mathbb{R}$
   • $\mathbb{C}$
   • $\mathbb{F}_2$
3. Find the eigenspaces of the matrix $A$ over each field from Problem 2.