Fundamental Exercises in Algebra

In studying for the Algebra Qualifying Exam, these are some exercises you should really really know. Most are either common questions on past qualifying exams or popular homework problems chosen from Hungerford’s Algebra.

Since these exercises are so fundamental, solutions to many of them can be found either in John Dusel’s notes, or in Kayla Murray’s notes, or somewhere online like Math Stack Exchange (MathSE). If you find a solution online, you should send me a link so I can post it here. Otherwise, if you think it’ll help you study, you can type up a solution and send me a PDF to post here. Or better, you can type up your solution on MathSE so that other algebra students can easily find it, add to it, comment on it, etc. The MathSE community is going through a bit of a phase right now, though, so it would be a good idea to read over this brief guide to posting on MathSE before writing up your solution there.

Linear Algebra

1. For a division ring $D$, let $V_i$ be a finite dimensional vector space over $D$ for $i \in \{1, \dotsc, k\}$. Suppose the sequence $$0 \longrightarrow V_1 \longrightarrow V_2 % \longrightarrow \dotsb \longrightarrow V_k \longrightarrow 0$$ is exact. Prove that $\sum_{i=1}^k (-1)^i \dim_D V_i = 0$.
2. Prove that if $A$ and $B$ are invertible matrices over a field $\boldsymbol{k}$, then $A+\lambda B$ is invertible for all but finitely many $\lambda \in \boldsymbol{k}$.
3. For the ring of $n \times n$ matrices over a commutative unital ring $R$, which we’ll denote $\mathrm{Mat}_n(R)$, recall the definition of the determinant map $\mathrm{det}\colon \mathrm{Mat}_n(R) \to R$. For $A \in \mathrm{Mat}_n(R)$ also recall the definition of the classical adjoint $A^a$ of $A$. Prove that:
   1. $\mathrm{det}\left(A^a\right) = \mathrm{det}(A)^{n-1}$
   2. $\left(A^a\right)^a = \mathrm{det}(A)^{n-2} A$
4. If $R$ is an integral domain and $A$ is an $n \times n$ matrix over $R$, prove that if a system of linear equations $Ax = 0$ has a nonzero solution then $\mathrm{det}A = 0$. Is the converse true? What if we drop the assumption that $R$ is an integral domain?
5. What is the companion matrix $M$ of the polynomial $f = x^2 -x+2$ over $\boldsymbol{C}$? Prove that $f$ is the minimal polynomial of $M$.
6. Suppose that $\phi$ and $\psi$ are commuting endomorphisms of a finite dimensional vector space $E$ over a field $\boldsymbol{k}$, so $\phi\psi = \psi\phi$.
   1. Prove that if $\boldsymbol{k}$ is algebraically closed, then $\phi$ and $\psi$ have a common eigenvector.
   2. Prove that if $E$ has a basis consisting of eigenvectors of $\phi$ and $E$ has a basis consisting of eigenvectors of $\psi$, then $E$ has a basis consisting of vectors that are eigenvectors for both $\phi$ and $\psi$ simultaneously.