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($A$).
Let $A$ be a $n\times m$ matrix and $B$ be a $m\times n$ matrix such that $AB=I$ then show that $m\ge n$.
Let $V$ be a vector space, $\{v_1,\ldots, v_n\}$ and $\{w_1,\ldots,w_{n+1}\}$ be linearly independent subsets. Show that for some $i$ $\{w_i, v_1, \ldots, v_n\}$ is linearly independent. Conclude that in a finite dimensional vector space two bases have same size.
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 5, Exercise 1.2, 1.3, 1.4.
Artin Edition 2, Chapter 5, Exercise 3.1, 4.2, 4.6.
Artin Edition 2, Chapter 8, Exercise 2.1, 2.2.
Let $W\subset V$ be a vector subspace over a field $F$ (of characteristic possibly 2). Let $<,>$ be a symmetric bilinear form on $V$ whose restriction on $W$ is non-degenerate. Show that $V=W\oplus W^{\bot}$. Hint: Use the bilinear form to define a linear operator on $V$ whose kernel is $W^{\bot}$ and whose image is $W$.
Artin Edition 2, Chapter 8, Exercise 3.2, 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
Let $0\to A\to B \to C\to 0$ be a short exact sequence of $R$-modules. Show that if $A$ and $C$ are finitely generated then so is $B$.
Let $\mathscr{C}$ be an abelian category and let $f\in Mor(A,B)$ for some objects $A$ and $B$ of $\mathscr{C}$. Show that kernel($f$) is a monomorphism. Write down the universal property an object $I$ together with $i\in Mor(I,B)$ should satisfy for it to be image of $f$. Show that kernel(cokernel($f$)) satisfy this universal property.
Let $R$ be a ring and $M$, $N$ and $K$ be $R$-modules. Show that $Hom(M,N\oplus K)$ is isomorphic to $Hom(M,N)\oplus Hom(M,K)$ as $R$-modules.
Let $M$ be an $R$-module. Show that TFAE:
(i) Every submodule of $M$ is finitely generated.
(ii) Every increasing chain of submodules of $M$ is eventually constant.
(iii) Every nonempty collection of submodules of $M$ has a maximal element with respect to inclusion.
Let $0\to A\to B\to C\to 0$ be an exact sequence of $R$-modules. Show that if $A$ and $C$ are noetherian $R$-module then so is $B$.
Let $R$ be an integral domain and $M$ an $R$-module. Let $N$ be an $R$-submodule of $M$. Show that rank$(M)=$ rank$(N) +$ rank$(M/N)$.
Dummit and Foote, Chapter 12, Section 12.2; 4, 11, 18
Let $a(x)\in k[x]$ be a nonconstant monic polynomial. Show that the minimal polynomial and the characteristic polynomial of the companion matrix $C(a)$ is $a(x)$.
Let $A$ be a $n\times n$ matrix over a PID $R$. Show that for each $1\le i \le n$, the $i^{th}$ diagonal entries of any two Smith normal form of $A$ are associates.
Let $A$ be a $n\times n$ matrix over a field $k$. Let $R=k[x]$. Consider $V=k^n$ as a $R$-module via $x\cdot v:=Av$ for $v\in V$. Show that $V$ is cokernel of the $R$-linear map $\theta:R^n\to R^n$ given by the matrix $xI-A$.
Dummit and Foote, Chapter 12, Section 12.3; 7, 25.