Homework Algebra II, B.Math

Week 1

1. Artin, Chapter 1, Section 1, Exercise 5, 10, 11.

2. Artin, Chapter 1, Section 2, Exercise 2.

   Week 2

3. Artin Edition 2, Chapter 1, Exercise 2.8, 2.9

4. Artin Edition 2, Chapter 1, Exercise 3.1, 3.2

5. Let $E$ be an elementary matrix and $A$ be a matrix. Show $AE$ is a matrix obtained from $A$ by the analogous coloumn operation associated to $E$. That is if $E$ is such that $EA$ has $i^{th}$ and $j^{th}$ row of $A$ switched then $AE$ has $i^{th}$ and $j^{th}$ coloumn switched and so on.

   Week 3

6. Show that expansion by 1st row formula for determinant holds.

7. Artin Edition 2, Chapter 1, Exercise 4.6, M1

8. Artin Edition 2, Chapter 1, Exercise M10.

9. Artin Edition 2, Chapter 3, Exercise 1.1, 1.5.

10. Artin Edition 2, Chapter 3, Exercise 2.2.

    Week 4

11. Let $V$ be a vector space and $S, S'$ be subsets of $V$. If $S\subset$ span$(S')$ then span$(S)\subset$ span$(S')$

12. Let $A$ be a $n\times m$ matrix and $B$ be a $m\times n$ matrix such that $AB=I$ then show that $m\ge n$.

13. Artin Edition 2, Chapter 3, Exercise 3.1, 3.3, 3.4, 3.6

    Week 5

14. Let $A$ be a $m\times n$ matrix. Show that coloumn rank of $A$ and row rank of $A$ are equal.

    Week 6

15. Artin Edition 2, Chapter 3, Exercise 4.5, 5.1, 5.2

    Week 7

16. Artin Edition 2, Chapter 4, Exercise 1.4, 2.2, 2.3, 3.4, 4.3, 4.6, 5.10, 6.8

17. Let $\phi$ and $\psi$ be endomorphisms of a finite dimensional vector space $V$ over the field $F$. Let $\mathscr{B}$ be an ordered basis of $V$. Let $A$ and $B$ be the matrices of $\phi$ and $\psi$ w.r.t. $\mathscr{B}$. Show that $A+B$ and $AB$ are matrices of $\phi+\psi$ and $\phi\circ\psi$ w.r.t. $\mathscr{B}$. Conclude that $f(A)$ is the matrix of $f(\phi)$ w.r.t. $\mathscr B$ for any polynomial $f(x)\in F[x]$.

    Week 8 and 9

18. Artin Edition 2, Chapter 4, Exercise 7.2,

19. Let $A$ and $B$ be square matrices (not necessarily invertible). Show that characteristic polynomial of $AB$ and $BA$ are same.

20. Artin Edition 2, Chapter 5, Exercise 1.2, 1.3, 1.4, 3.1, 3.3b,d, 4.2, 4.6

21. Artin Edition 2, Chapter 8, Exercise 2.1, 2.2

    Week 10

22. Artin Edition 2, Chapter 8, Exercise 3.1, 3.2, 3.5

23. Artin Edition 2, Chapter 8, Exercise 4.1, 4.4, 4.9, 4.11

24. Artin Edition 2, Chapter 8, Exercise 5.5, 5.6

    Week 11

25. Let $T$ be a diagonalizable linear operator on a vector space $V$ and $W$ be a $T$-invariant subspace. Show that $T$ restricted to $W$ is diagonalizable.

26. Artin Edition 2, Chapter 8, Exercise 6.5, 6.7, 6.13, 6.15, 6.17, 7.1