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数学代写|现代代数代考Modern Algebra代写|Permutations and Inverses

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数学代写|现代代数代考Modern Algebra代写|Permutations and Inverses

A one-to-one correspondence from a set $A$ to itself is called a permutation on $A$. For any nonempty set $A$, we adopt the notation $\mathcal{S}(A)$ as standard for the set of all permutations on $A$. The set of all mappings from $A$ to $A$ will be denoted by $\mathcal{M}(A)$.

From the discussion at the end of Section 1.2, we know that composition of mappings is an associative binary operation on $\mathcal{M}(A)$. The identity mapping $I_A$ is defined by
$$I_A(x)=x \text { for all } x \in A .$$
For any $f$ in $\mathcal{M}(A)$,
$$\left(I_A \circ f\right)(x)=I_A(f(x))=f(x)$$
and
$$\left(f \circ I_A\right)(x)=f\left(I_A(x)\right)=f(x)$$
so $I_A \circ f=f \circ I_A=f$. That is, $I_A$ is an identity element for mapping composition. Once an identity element is established for a binary operation, the next natural question is whether inverses exist. Consider the mappings in the next example.

Example 1 In Example 1 of Section 1.3, we defined the mappings $f: \mathbf{Z} \rightarrow \mathbf{Z}$ and $g: \mathbf{Z} \rightarrow \mathbf{Z}$ by
$$f(n)=2 n$$
and
$$g(n)= \begin{cases}\frac{n}{2} & \text { if } n \text { is even } \ 4 & \text { if } n \text { is odd. }\end{cases}$$
For these mappings, $(g \circ f)(n)=n$ for all $n \in \mathbf{Z}$, so $g \circ f=I_Z$ and $g$ is a left inverse for $f$. Note, however, that
$$(f \circ g)(n)= \begin{cases}n & \text { if } n \text { is even } \ 8 & \text { if } n \text { is odd }\end{cases}$$

数学代写|现代代数代考Modern Algebra代写|Matrices

The word matrix is used in mathematics to denote a rectangular array of elements in rows and columns. The elements in the array are usually numbers, and brackets may be used to mark the beginning and the end of the array. Two illustrations of this type of matrix are
$$\left[\begin{array}{rrrr} 5 & -1 & 0 & 3 \ 2 & 1 & -2 & 7 \ 4 & -6 & 4 & 3 \end{array}\right] \text { and }\left[\begin{array}{rr} 9 & 1 \ -1 & 0 \ 6 & -3 \end{array}\right]$$
The formal notation for a matrix is introduced in the following definition. We shall soon see that this notation is extremely useful in proving certain facts about matrices.
An $\boldsymbol{m}$ by $\boldsymbol{n}$ matrix over a set $S$ is a rectangular array of elements of $S$, arranged in $m$ rows and $n$ columns. It is customary to write an $m$ by $n$ matrix using notation such as
$$A=\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \ a_{21} & a_{22} & \cdots & a_{2 n} \ \vdots & \vdots & & \vdots \ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end{array}\right],$$
where the uppercase letter $A$ denotes the matrix and the lowercase $a_{i j}$ denotes the element in row $i$ and column $j$ of the matrix $A$. The rows are numbered from the top down, and the columns are numbered from left to right. The matrix $A$ is referred to as a matrix of dimen$\operatorname{sion} m \times n$ (read ” $m$ by $n$ “).

现代代数代写

数学代写|现代代数代考Modern Algebra代写|Permutations and Inverses

$$I_A(x)=x \text { for all } x \in A .$$

$$\left(I_A \circ f\right)(x)=I_A(f(x))=f(x)$$

$$\left(f \circ I_A\right)(x)=f\left(I_A(x)\right)=f(x)$$

$$f(n)=2 n$$

$$g(n)= \begin{cases}\frac{n}{2} & \text { if } n \text { is even } \ 4 & \text { if } n \text { is odd. }\end{cases}$$

$$(f \circ g)(n)= \begin{cases}n & \text { if } n \text { is even } \ 8 & \text { if } n \text { is odd }\end{cases}$$

数学代写|现代代数代考Modern Algebra代写|Matrices

“矩阵”一词在数学中用来表示按行和列排列的元素的矩形数组。数组中的元素通常是数字，括号可以用来标记数组的开始和结束。这类矩阵的两个例子是
$$\left[\begin{array}{rrrr} 5 & -1 & 0 & 3 \ 2 & 1 & -2 & 7 \ 4 & -6 & 4 & 3 \end{array}\right] \text { and }\left[\begin{array}{rr} 9 & 1 \ -1 & 0 \ 6 & -3 \end{array}\right]$$

$$A=\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \ a_{21} & a_{22} & \cdots & a_{2 n} \ \vdots & \vdots & & \vdots \ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end{array}\right],$$