\(F\) denotes an algebraically closed field of characteristic 0. If you like, you can suppose that \(F = \mathbf{C}\) is the field of complex numbers.
Let \(V\) be a finite dimensional vector space over the field \(F\). Suppose that \(\phi,\psi:V \to V\) are linear maps. Let \(\lambda \in F\) be an eigenvalue of \(\phi\) and write \(W\) for the \(\lambda\)-eigenspace of \(\phi\); i.e. \[W = \{v \in V \mid \phi(v) = \lambda v \}.\] If \(\phi \psi = \psi \phi\) show that \(W\) is invariant under \(\psi\) – i.e. show that \(\psi(W) \subseteq W\).
Let \(n \in \mathbf{N}\) be a non-zero natural number, and let \(V\) be an \(n\) dimensional \(F\)-vector space with a given basis \(e_1,e_2,\cdots,e_n\).
Consider the linear transformation \(T:V \to V\) given by the rule \[Te_i = e_{i+1 \pmod n}.\] In other words \[Te_i = \left \{ \begin{matrix} e_{i+1} & i < n \\ e_1 & i =n \end{matrix} \right ..\]
Show that \(T\) is invertible and that \(T^n = \operatorname{id}_V\).
Consider the vector \(v_0 = \displaystyle \sum_{i=1}^n e_i\). Show that \(v_0\) is a \(1\)-eigenvector for \(T\).
Let \(\zeta \in F\) be a primitive \(n\)-th root of unity. (e.g. if you assume \(F = \mathbf{C}\), you may as well take \(\zeta = e^{2\pi i/n}\)).
Let \(v_1 = \displaystyle \sum_{i=1}^n \zeta^i e_i\). Show that \(v_1\) is a \(\zeta^{-1}\)-eigenvector for \(T\).
More generally, let \(0 \le j < n\) and let \[v_j = \sum_{i=1}^n \zeta^{ij} e_i.\] Show that \(v_j\) is a \(\zeta^{-j}\)-eigenvector for \(T\).
Conclude that \(v_0,v_1,\cdots,v_{n-1}\) is a basis of \(V\) consisting of eigenvectors for \(T\), so that \(T\) is diagonalizable.
Hint: You need to use the fact that eigenvectors for distinct eigenvalues are linearly independent.
What is the matrix of \(T\) in this basis?
Let \(G = \mathbb{Z}/3\mathbb{Z}\) be the additive group of order \(3\), and let \(\zeta\) be a primitive \(3\)rd root of unity in \(F\).
To define a representation \(\rho:G \to \operatorname{GL}_n(F)\), it is enough to find a matrix \(M \in \operatorname{GL}_n(F)\) with \(M^3 = 1\); in turn, \(M\) determines a representation \(\rho\) by the rule \(\rho(i + 3\mathbb{Z}) = M^i\).
Consider the representation \(\rho_1 : G \to \operatorname{GL}_3(F)\) given by the matrix \[\rho_1(1 + 3\mathbb{Z}) = M_1 = \begin{bmatrix} 1 & 0 & 0\\ 0 & \zeta & 0 \\ 0 & 0 & \zeta^2 \end{bmatrix}\] and consider the representation \(\rho_2:G \to \operatorname{GL}_3(F)\) given by the matrix \[\rho_2(1 + 3\mathbb{Z}) = M_2 = \begin{bmatrix} 0 & 0 & 1\\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}.\]
Show that the representations \(\rho_1\) and \(\rho_2\) are equivalent (alternative terminology: are isomorphic). In other words, find a linear bijection \(\Phi:F^3 \to F^3\) with the property that \[\Phi(\rho_2(g)v) = \rho_1(g)\Phi(v)\] for every \(g \in G\) and \(v \in F^3\).
Hint: First find a basis of \(F^3\) consisting of eigenvectors for the matrix \(M_2\).
Let \(V\) be a \(n\) dimensional \(F\)-vector space for \(n \in \mathbb{N}\).
Let \(\operatorname{GL}(V)\) denote the group \[\operatorname{GL}(V) = \{ \text{all invertible $F$-linear transformations $\phi:V \to V$}\}\] where the group operation is composition of linear transformations.
Recall that \(\operatorname{GL}_n(F)\) denotes the group of all invertible \(n \times n\) matrices.
If \(\mathcal{B} = \{b_1,b_2,\cdots,b_n\}\) is a choice of basis, show that the assignment \(\phi \mapsto [\phi]_{\mathcal{B}}\) determines an isomorphism \[\operatorname{GL}(V) \xrightarrow{\sim} \operatorname{GL}_n(F).\]
Here \([\phi]_{\mathcal{B}} = [M_{ij}]\) denotes the matrix of \(\phi\) in the basis \(\mathcal{B}\) defined by equations
\[\phi(b_i) = \sum_{k=1}^n M_{ki} b_k.\]