Math 520B
Written Assignment No. 5

due Friday, December 9, 2005

Directions. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises is not permitted. You may not seek help from others.

1. Let $f$ be the $\mathbf{Z}$-linear map from ${\mathbf{Z}}^4$ to ${\mathbf{Z}}^3$ defined by $f\left(x\right)=Mx$ for all $x\in {\mathbf{Z}}^4$ where M is the $3\times 4$ matrix $$M=\left(\begin{array}{cccc}24& 30& -36& -42\\ 14& 24& -34& -44\\ 34& 48& -62& -76\end{array}\right)\text{.}$$ Represent the cokernel of $f$ as a direct sum of cyclic groups.

2. Find one matrix in each of the similarity classes of matrices over the field $\mathbf{C}$ (of complex numbers) that share the characteristic polynomial $t^3-t^2-t+1$.

3. Let $A$ be the $5\times 5$ matrix in the field ${\mathbf{F}}_2=\mathbf{Z}/2\mathbf{Z}$ $$A=\left(\begin{array}{ccccc}1& 0& 0& 0& 1\\ 0& 1& 1& 1& 1\\ 0& 1& 0& 1& 0\\ 1& 0& 1& 1& 1\\ 0& 1& 1& 0& 0\end{array}\right)\text{.}$$ Find a direct sum of companion matrices that is similar to $A$.

4. Let $M$ be the $6\times 6$ rational matrix $$M=\left(\begin{array}{cccccc}102& -52& 71& -21& 16& -16\\ 324& -157& 228& -69& 78& -57\\ 357& -182& 244& -66& 49& -60\\ 532& -286& 356& -88& 19& -82\\ 36& -26& 20& 3& -24& -6\\ 532& -286& 356& -90& 19& -80\end{array}\right)\text{.}$$ Find the sequence of successively divisible invariant factors as well as the minimal and characteristic polynomials of $M$.

5. Let $V$ and $W$ be $n$-dimensional vector spaces over a field $F$, and let $b:V\times W\to F$ be $F$-bilinear. The bilinear form $b$ determines a linear map ${\varphi }_b\in {\mathrm{Hom}}_F\left(V,\check{W}\right)$, with $\check{W}={\mathrm{Hom}}_F\left(W,F\right)$, that is defined by $${\varphi }_b\left(v\right)=\left[w\mapsto b\left(v,w\right)\right]\text{.}$$

   1. Show that ${\varphi }_b$ is an isomorphism if and only if for given bases $v_1,\dots ,v_n$ of $V$ and $w_1,\dots w_n$ of $W$ one has $\det \left(b\left(v_i,w_j\right)\right)\ne 0$.

   2. Produce such a bilinear form in the case where $V={\Lambda }^{p}U$ and $W={\Lambda }^{k-p}U$ when $U$ is a $k$-dimensional vector space over $F$.

   3. To what extent may one generalize the two preceding items to correct statements when $F$ is a commutative ring, $V,W$ free $F$-modules of rank $n$, and $U$ a free $F$-module of rank $k$?