MA251 - Algebra I

Assignment 4 November 2009

Answer the questions on your own paper. Write your own name in the top left-hand corner, and your university ID number in the top right-hand corner. Use the problems at the beginning as well as exercises in the lecture notes for a warm up. Solutions to the FOUR TEST problems must be handed in by 15.00 on MONDAY 30 NOVEMBER (Monday of the ninth week of term), or they will not be marked. There will be an award of 5 extra marks for clarity, so do a good job.

These are practice problems for you to sharpen your teeth on.

P1. Let $A$ be a symmetric $n\times n$ real matrix. We consider a function $f:S \rightarrow \mathbb{R}$ where $f (\ensuremath{\mathbf{v}}\xspace ) = \ensuremath{\mathbf{v}}\xspace ^T A \ensuremath{\mathbf{v}}\xspace$ and $S=\{\ensuremath{\mathbf{v}}\xspace \in\mathbb{R}^n \; \vert \; \ensuremath{\mathbf{v}}\xspace ^T\ensuremath{\mathbf{v}}\xspace =1\}$ is the unit sphere. The set $S$ is compact and the function $f$ is continuous. It is known in Analysis (Extreme Value Theorem or Weierstrass' Theorem) that there exists a point $\ensuremath{\mathbf{v}}\xspace _0\in S$ such that $f(\ensuremath{\mathbf{v}}\xspace )$ achieves its maximum at $\ensuremath{\mathbf{v}}\xspace =\ensuremath{\mathbf{v}}\xspace _0$. Prove that $\ensuremath{\mathbf{v}}\xspace _0$ is an eigenvector of $A$.

(Note: This is another proof of the fact that a symmetric matrix admits a real eigenvector, using methods of Analysis)

P3. Prove that elements $x_1 , . . . , x_r \in \mathbb{Z}^n$ are linearly independent if and only if they are linearly independent over $\mathbb{Q}$ when regarded as vectors in $\mathbb{Q}^n$ .

P4. For the following finitely generated abelian groups $G$, write down their corresponding matrix, reduce it to Smith Normal Form, and hence express $G$ as a direct sum of cyclic groups:

(i) $\langle x_1 , x_2 , x_3 \vert x_1 - 2x_2 , x_1 + 6x_2 + 8x_3 \rangle$;

(ii) $\langle x_1 , x_2 \vert 6 x_1 - 6 x_2 , -6 x_1 -12 x_2 , 4x_1- 8x_2 \rangle$ .

P5. Write down the possible isomorphism types of abelian groups of orders 74, 147, 666, 800 and 1221.

P6. Show that the Smith Normal Form of a unimodular matrix with entries in $\mathbb{Z}$ is the identity matrix. Deduce that any such unimodular matrix can be expressed as the product of elementary unimodular matrices.

P7. Find all subgroups of the group $\mathbb{Z}_2 \oplus \mathbb{Z}_4$. (There are eight altogether.) Express each subgroup as a direct sum of cyclic groups.

P8. Proof of the uniqueness of the Smith Normal Form. Let $A\in \mathbb{Z}^{m\times n}$ be an $m \times n$ matrix with entries in $\mathbb{Z}$. For $1 \leq i \leq \min(m, n)$, an $i \times i$-submatrix of A is defined to be a matrix obtained from A by deleting any $m-i$ rows and any $n-i$ columns of $A$. Define

$$\gamma_i (A) = \mbox{gcd}(\{ \vert \det(S)\vert \; ; \; S \mbox{ is an } i \times i-\mbox{submatrix of } A \}).$$

(The convention here is that $\mbox{gcd}(0, n) = n$ for any $n \geq 0$.)

(i) Show that, if $B$ is obtained from $A$ by applying elementary unimodular row and column operations, then $\gamma_i (B) = \gamma_i (A)$ for $1 \leq i \leq \min(m, n)$. (This is easy for (UR2), (UR3), but a little harder for (UR1).)

(ii) Show that, if $B$ is Smith Normal Form with nonzero diagonal entries $\alpha_1 , . . . , \alpha_r$ , then $\gamma_i (B) = \alpha_1 \alpha_2 \ldots \alpha_i$ for $1 \leq i \leq r$ and $\gamma_i (B) = 0$ for all $i > r$.

(iii) Deduce that the Smith Normal Form is uniquely determined by A.

P9. A group is a set with a binary operation that satisfies all axioms of an Abelian group except commutativity. Homomorphism between groups is a function $f$ satisfying $f(x+y)=f(x)+f(y)$.

(i) Prove that a group $G$ is abelian if and only if $f:G\rightarrow G$ defined by $f(x)=2x$ is a group homomorphism.

(ii) Prove that if a group $G$ satisfies the property that $2g=1$ for all $g \in G$ then $G$ is abelian.

The following problems are test problems for you to submit for marking. Write concise but complete solutions only to the questions asked. Additional 5 marks are awarded for clarity.

1. (i) Find the orders of all elements in the group $\mathbb{Z}_{12}$. [1 mark]

(ii) Write down and prove a formula for the order of an element $k\in\mathbb{Z}_n$ for general $n$ and $k$. [2 marks]

2. Let $G$ be an abelian group and let $g, h \in G$.

(i) If $\vert g\vert = m$ is finite then prove that, for $n\in \mathbb{Z}, ng = 0$ if and only if $m\vert n$. [2 marks]

(ii) Let us assume that $\vert g\vert$ and $\vert h\vert$ are both finite, with $\mbox{hcf}(\vert g\vert, \vert h\vert) = 1$. Prove that $\vert g + h\vert = \vert g\vert\vert h\vert$. [2 marks]

(iv) Prove that $\mathbb{Z}_m \oplus \mathbb{Z}_n \cong \mathbb{Z}_{mn}$ if and only if $m$ and $n$ are relatively prime. [2 marks]

3. We are working with a finite dimensional Euclidean space $(V,\; <,> \;)$. Given vectors $\ensuremath{\mathbf{v}}\xspace _1, \ldots \ensuremath{\mathbf{v}}\xspace _n\in V$, one defines the parallelepiped

$$P(\ensuremath{\mathbf{v}}\xspace _1, \ldots \ensuremath{\mat... ...\mathbf{v}}\xspace _i \in V \; \vert \; \alpha_i \in [0,1] \}.$$

The $n$-dimensional volume of this parallelepiped is defined inductively:

$${{\mathrm{Vol}}}_1(P(\ensuremath{\mathbf{v}}\xspace _1))= \ve... ...e _{n-1}))) \vert\vert\ensuremath{\mathbf{w}}\xspace \vert\vert$$

where $\ensuremath{\mathbf{w}}\xspace =\ensuremath{\mathbf{v}}\xspace _n+\alpha_1 \ens... ...thbf{v}}\xspace _1 + \ldots + \alpha_{n-1}\ensuremath{\mathbf{v}}\xspace _{n-1}$ is the unique¹vector of this form orthogonal to all $\ensuremath{\mathbf{v}}\xspace _i$ for $i\leq n-1$. Intuitively, we define the $n$-dimensional volume as a product of the $n-1$-dimensional volume of a base and the height. There is nothing to prove at this point.

(i) Using the Gram matrix from HW-3, prove that

$$\det (G(\ensuremath{\mathbf{v}}\xspace _1, \ldots \ensuremath... ...bf{v}}\xspace _1, \ldots \ensuremath{\mathbf{v}}\xspace _n))^2.$$

[2 marks]

(ii) Now let $n$ be the dimension of $V$. Let $A$ be a square matrix whose columns are $\underline{\ensuremath{\mathbf{v}}\xspace }_i$, the coordinate vectors of $\ensuremath{\mathbf{v}}\xspace _i$ in some orthonormal basis. Using part (i), prove that

$$\vert \det (A) \vert = {{\mathrm{Vol}}}_n(P(\ensuremath{\mathbf{v}}\xspace _1, \ldots \ensuremath{\mathbf{v}}\xspace _n)).$$

[2 marks]

4. We consider finite dimensional vector spaces $U$ and $V$ over complex numbers $\mathbb{C}$, their bases $\ensuremath{\mathbf{e}}\xspace _i\in U$, $i=1,2\ldots n$ and $\ensuremath{\mathbf{f}}\xspace _i\in V$, $i=1,2\ldots m$, their dual spaces $U^\ast$ and $V^\ast$, and the dual bases $\ensuremath{\mathbf{e}}\xspace ^i$ and $\ensuremath{\mathbf{f}}\xspace ^i$.

(i) Let $T:U\rightarrow V$ be a linear map. We consider a function $T^\star : V^\ast \rightarrow U^\ast$ so that for each $\alpha\in V^\ast$, $T^\star (\alpha)$ is an element of $U^\ast$ defined by

$$T^\star (\alpha) \; (\ensuremath{\mathbf{u}}\xspace ) = \alp... ...pace )) \mbox{ for all } \ensuremath{\mathbf{u}}\xspace \in U .$$

Prove that $T^\star$ is a linear map. [2 marks]

The linear map $T^\star$ in (i) is called the dual linear map of $T$.

(ii) Suppose $A$ is the matrix of $T$ and $B$ is the matrix of $T^\star$ in the bases described above. Prove that $B=A^T$. [2 marks]

(iii) Now we assume that both vector spaces are Hermitian. As in the Problem 2, HW-1 we consider $T_U:U \rightarrow U^\ast$ defined by

$$T_U(\ensuremath{\mathbf{w}}\xspace )(\ensuremath{\mathbf{u}}... ...{\mathbf{w}}\xspace , \ensuremath{\mathbf{u}}\xspace \in U.$$

Prove that if the basis $\ensuremath{\mathbf{e}}\xspace _i$ is orthonormal them $T_U(\ensuremath{\mathbf{e}}\xspace _i) = \ensuremath{\mathbf{e}}\xspace ^i$. [1 marks]

It follows from (iii) that $T_U$ is an anti-linear²bijection between $U$ and $U^\ast$. Hence, we can write its inverse in the following part (iv).

(iv) Two linear maps $T:U\rightarrow V$ and $S:V\rightarrow U$ are formally dual if

$$<T(\ensuremath{\mathbf{u}}\xspace ),\ensuremath{\mathbf{v}}\... ...hbf{v}}\xspace \in V, \; \ensuremath{\mathbf{u}}\xspace \in U.$$

Prove that the linear³ maps $T$ and $T_U^{-1} T^\star T_V$ are formally dual. [2 marks]

(Hint: Compute the matrices of both sesquilinear maps $U\times V \rightarrow \mathbb{C}$ in a pair of orthonormal bases. )