課程名稱︰代數一
課程性質︰數學系選修 數學系碩士班數學組必選
課程教師︰朱樺
開課系所︰數學系所
考試時間︰941124
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(1) (25%)
(a) Let $G$ be a group and $H$ be a subgroup. Prove or disprove:
$Z(H) \subset Z(G)$.
(b) Find the center of $\SL_n(\mathbb{Z})$.
(c) Find the order of $Z(\SL_n(\mathbb{Z}))$.
(2) (30%)
(a) How many subspaces does $V=(\mathbb{F}_p)^5$ have?
(b) Let $V$ be a vector space over a field $F$ of dimension > 1.
Define $k=k(V)$ to be the least integer such that $V$ is a union of
$k$ proper subspaces. Show that $k(\mathbb{F}_p^2)=p+1$.
(c) Find $k(\mathbb{F}_2^3)$ and $k(\mathbb{F}_3^3)$. Give a conjecture
for $k(\mathbb{F}_p^3)$.
(d) Can you give a conjectture for $k(\mathbb{F}_p^n)$?
(e) Show that, if $F$ is an infinite field, then $k(V)=\ifnty$.
(3) (30%)
(a) Let $F$ be a field and $A$, $B \in M_n(\mathbb{F})$. Show that, if
one if $A$ and $B$ is invertible, then $AB$ and $BA$ are similar.
(b) What is wrong with the following argument?
Let $A=[a_{ij}]$ and $B=[b_{ij}] \in M_n(\mathbb{F})$. Define
$\tilde{A}=[x_{ij}]$ and $\tilde{B}=[y_{ij}] \in M_n(K)$, where
$K=\mathbb{Q}(x_{ij},y_{ij})$ is the rational function field of $2n^2$
variables. Since $\det \tilde{A} \neq 0$, it is invertible. Thus by
(a), $\tilde{A}\tilde{B}$ and $\tilde{B}\tilde{A}$ are similar. That
is, there is a matrix $P$ such that $P\tilde{A}\tilde{B}P^{-1}
=\tilde{B}\tilde{A}$.(*)
Now define a ring homomorphism $\Phi: \mathbb{Z}(x_{ij},y_{ij})
\rightarrow F$, by $x_{ij} \mapsto a_{ij}$, $y_{ij} \mapsto b{ij}$.
Then $\Phi(\tilde{A})=A$, $\Phi(\tilde{B})=B$. Apply $\Phi$ to (*)
and we get that $AB$ and $BA$ are similar.
(c) Prove or disprove: for any two matrices $A$, $B \in M_n(\mathbb{F})$,
$AB$ and $BA$ are similar.
(d) Prove that $AB$ and $BA$ have the same characteristic polynomial.
(4) (25%)
(a) If $A$ is a real square matrix, then $A^tA$ has only nonegative
eigenvalues.
(b) If $A$ is a real square matrix, then $A=QB$ for some symmetric matrix
$B$ and orthgonal matrix $Q$.
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