Let $A$ be a $n\times n$ matrix with entries in a field $F$. Let $v_1,\ldots v_n$ be coloumn vectors of $A$.
(a)Show that det($A$) is a $n$-multi-linear function in coloumn vectors of $A$, i.e., for all $1\le i \le n$ det($(v_1,\ldots,v_{i-1}, av+v',v_{i+1},\ldots,v_n)$)=$a$det($(v_1,\ldots,v_{i-1}, v,v_{i+1},\ldots,v_n)$)+det($(v_1,\ldots,v_{i-1},v',v_{i+1},\ldots,v_n)$) for all $a\in F$ and $v,v'\in F^n$.
(b)det($(v_1,\ldots,\ldots,v_n)$)=0 if $v_i=v_{i+1}$ for some $i$ and det($I$)=1.
(c)Show that converse holds. That is a function on $n\times n$ matrix satisfying (a) and (b) is same as det($A$).
Artin Edition 2, Chapter 3, Exercise 2.2, 3.1, 3.3, 3.4, 4.5, 5.1, 5.2
Artin Edition 2, Chapter 4, Exercise 1.4, 2.3, 3.4, 4.6, 5.10, 6.8
Let $\phi$ be a linear operator on a vector space $V$ and $W$ be an invariant subspace of $\phi$. Let $\psi$ be the restriction $\phi$ to $W$. Show that the characteristic (respectively minimal) polynomial of $W$ divides characteristic (respectively minimal) polynomial of $V$.
Let $A$ and $B$ be square matrices (not necessarily invertible). Show that characteristic polynomial of $AB$ and $BA$ are same.
Artin Edition 2, Chapter 4, Exercise 7.2, 7.4
Let $\phi$ be a linear operator on a finite dimensional vector space $V$ and $\lambda$ be an eigen value of $\phi$. Let $V_{\lambda}$ be the generalized eigen subspace of $V$ and $\phi'$ be the restriction of $\phi$ to $V_{\lambda}$. If the characteristic polynomial $ch_{\phi}(t)=(t-\lambda)^mg(t)$ where $g(\lambda)\ne 0$, show that $ch_{\phi'}(t)=(t-\lambda)^m$.
Let $F$ be a field and $f(x),g(x)$ be two polynomials with coefficient in $F$. Let $h(x)$ be a greatest common divisor of $f(x)$ and $g(x)$. Show thatthere exist polynomials $a(x)$ and $b(x)$ such that $a(x)f(x)+b(x)g(x)=h(x)$. (Hint: First do the case where $h(x)=1$ using Remainder's theorem. Deduce the general case from the $h(x)=1$ result.)
Artin Edition 2, Chapter 4, Exercise 7.5, 7.6, 7.8, 7.9
Let $A$ be a square matrix over complex numbers such that its characteristic and minimal polynomial are same. What are the possibilities for the Jordan form of $A$ in terms of its eigen values.
Prove the second isomorphism theorem for vector spaces.
Artin Edition 2, Chapter 5, Exercise 1.2, 1.3, 1.4.
Artin Edition 2, Chapter 5, Exercise 3.1, 4.2, 4.6.
Artin Edition 2, Chapter 6, Exercise 3.2, 3.3, 3.6.
Artin Edition 2, Chapter 6, Exercise 4.3, 5.1, 5.4, 7.2, 7.9.
Artin Edition 2, Chapter 6, Exercise 10.3, 11.1, 11.6, 11.8.
Artin Edition 2, Chapter 7, Exercise 2.2, 2.3, 2.11, 3.2.
Show that a group of order $pqr$ is not simple where $p$, $q$ and $r$ are primes.
Show that all the composition factors of a finite abelian group are of prime order.
Show that if $|G|=231$ then 11-sylow and 7-sylow subgroups are normal and $Z(G)$ contains the 11-sylow subgroup.
Artin Edition 2, Chapter 8, Exercise 2.1, 2.2.
Artin Edition 2, Chapter 8, Exercise 3.3, 3.5, 4.9, 4.11, 5.5, 5.6
Artin Edition 2, Chapter 8, Exercise 6.5, 6.7, 6.13, 6.15
Artin Edition 2, Chapter 8, Exercise 7.1, 7.4, 8.1, 8.2