User:J D Bowen/Math725 HW13

1) Given an operator $$T:V\to V \ $$, let $$B \ $$ be the Jordan basis of $$V \ $$ with respect to $$T \ $$. Then

$$\mathfrak{M}_B^B(T) = \begin{pmatrix} J_1 & \;    & \; \\ \; & \ddots & \; \\ \; & \;     & J_p\end{pmatrix}$$

where each block Ji is a square matrix of the form


 * $$J_i =

\begin{pmatrix} \lambda_i & 0          & \;     & 0  \\ 1       & \lambda_i    & \ddots & \;  \\ \;       & 1          & \ddots & 0  \\ 0       & \;           & 1     & \lambda_i \end{pmatrix},$$

and $$\text{dim}(J_i) = n_{\lambda_i} \ $$.

Hence $$\text{Tr}(T)= \Sigma \ \text{diagonal} = \Sigma n_{\lambda_i}\lambda_i \ $$.

2) Suppose $$\text{dim}(V)=n \ $$ and the characteristic polynomial of $$T \ $$ is $$c_T(x)=x^n+c_{n-1}x^{n-1}+\dots+c_0 \ $$.

We have $$c_T(x)=x^n+c_{n-1}x^{n-1}+\dots+c_0=\Pi_i (x-\lambda_i)^{n_{\lambda_i}} \ $$, but observe that in this product, terms of the power $$x^{n-1} \ $$ can only come from multiplying every single $$x \ $$ in the product except one, which is multiplied by $$-\lambda_i \ $$. Collecting all the powers of $$x^{n-1} \ $$, we see this is $$-\Sigma n_{\lambda_i}\lambda_i = -\text{Tr}(T) \ $$ by problem 1.

3) Given $$\left\{{0}\right\} = V_0 \subset \dots \subset V_i \subset \dots \subset V_t = V \ $$, observe that the projection map $$\pi_t:V_t\to V_t/V_{t-1} \ $$ has kernel $$V_{t-1} \ $$ and image V_t/V_{t-1} \.

Therefore, $$\text{dim}(V)=\text{dim}(V_{t-1})+\text{dim}(V_t/V_{t-1}) \ $$.

Now suppose that there exists an $$i, \ 1\leq i \leq t \ $$, such that $$\text{dim}(V)= \text{dim}(V_i)+\Sigma \text{dim}(V_j/V_{j-1}) \ $$. Then since the map $$\pi_i \ $$ has kernel $$V_i \ $$ and image $$V_i/V_{i-1} \ $$, and so $$\text{dim}(V)= \text{dim}(V_{i-1})+\Sigma \text{dim}(V_j/V_{j-1}) \ $$.

Notably, this means $$\text{dim}(V)=\text{dim}(V_1)+\Sigma \text{dim}(V_j/V_{j-1}) \ $$.

Since $$V_0 = \left\{{0}\right\}, \ V_1=V_1/V_0 \ $$ and so $$\text{dim}(V)=\Sigma \text{dim}(V_j/V_{j-1}) \ $$.

4)

5) Let $$V, W \ $$ be vector spaces, with $$U\subset V \ $$ a subspace and $$T:V\to W \ $$ linear, $$U\subset \text{ker}(T) \ $$. Define $$\pi:V\to V/U \ $$.