数学代写|线性代数代写linear algebra代考|AXIOMATIC DETERMINANT

This section is a bit more on the theoretical side than on the application side. It can be easily skipped, but some students may find it mathematically interesting. In it we determine the axioms for which the determinant is uniquely defined. One consequence is that all our definitions of determinant in earlier sections must coincide since they all satisfy these axioms. For the definition below, recall the notation for the cartesian product of vector spaces, namely for a vector space $V$,
$$V^n=\underbrace{V \times V \times \cdots \times V}_{\mathrm{n} \text { times }}=\left{\left(v_1, v_2, \ldots, v_n\right): v_1, v_2, \ldots, v_n \in V\right}$$
Definition 4.14 Let $V=\mathbb{R}^n$. A function $d: V^n \rightarrow \mathbb{R}$ is $n$-linear if it is linear in each of its cootdinates, i.e. for any $1 \leq i \leq n$ and any $v_1, \ldots, v_i, v_i^{\prime}, \ldots, v_n \in V$ we have

1. $d\left(v_1, \ldots, v_i+v_i^{\prime}, \ldots, v_n\right)=d\left(v_1, \ldots, v_i, \ldots, v_n\right)+d\left(v_1, \ldots, v_i^{\prime}, \ldots, v_n\right)$ and
2. $d\left(v_1, \ldots, a v_i, \ldots, v_n\right)=a d\left(v_1, \ldots, v_i, \ldots, v_n\right)$ for any $a \in \mathbb{R}$.

Example 4.33 We list here several important examples of $n$-linear functions.

1. Any linear transformation $T \in L\left(\mathbb{R}^n, \mathbb{R}\right)$ is a 1-linear function (these functions are sometimes called linear functionals).
2. Any inner product on $\mathbb{R}^n$ is a 2-linear (or bilinear) function.
3. If we represent a matrix $A=\left(c_1, c_2, \ldots, c_n\right)$ as an $n$-tuple of its columns, then the determinant is an $n$-linear function.

We list a couple results about $n$-linear functions which are left as exercises for the reader.
Lemma $4.8$

1. If $\left(v_1, v_2, \ldots, v_n\right)$ includes a coordinate which is the zero vector, then $d\left(v_1, v_2, \ldots, v_n\right)=0$.
2. Any linear combination of n-linear functions is again n-linear.
3. An n-linear function on $\mathbb{R}^n$ is completely determined by its values on the inputs
$$\left(e_{\sigma(1)}, e_{\sigma(2)}, \ldots, e_{\sigma(n)}\right),$$
where $e_1, e_2, \ldots, e_n$ is the standard basis for $\mathbb{R}^n$ and $\sigma$ is any permutation of the numbers $1,2, \ldots, n$.

数学代写|线性代数代写linear algebra代考|QUOTIENT VECTOR SPACE

If one is dealing with any algebraic structure, there is always a notion of a quotient structure. The reader should already have some good experience with equivalence relations and classes, otherwise it would be strongly recommended to study or review these concepts. In the first subsection of this section, we provide the reader with all the necessary review if needed. Quotient structures are important in the study of algebraic structures. Some important applications are the ability to equate algebraic structure via an isomorphism to a quotient structure. One important application we will see in this section is the First Isomorphism Theorem. Quotient structures can be useful in induction proofs where one has a notion of measuring size. In the case of a vector space, it is dimension which can be used for measuring size and therefore allows us to prove things by induction. One important application which is at the end of this section is the fact that every matrix is triagularizable over the complex numbers.

We begin with a review of equivalence relations which the reader may skip if they are already comfortable with this concept.

The notion of a relation on a set is important in many fields of mathematics. We shall see many applications of a particular type of relation (called an equivalence relation) in this text. We start by defining a relation and then narrow things down to an equivalence relation.

Definition $4.16 A$ relation $\sim$ on a set $A$ is simply any subset of the cartesian product $A \times A$. If $(a, b) \in \sim$ we instead write $a \sim b$ and we say a relates to $b$.

Example 4.35 Here, we list a number of examples including several that you have already seen in this text.

1. Let $A={a, b, c, d}$ and set $\sim={(a, b),(b, b),(c, d)}$. For instance, according to our definition of $\sim$, we have $c \sim d$ or $c$ relates to $d$.
2. Let $A=\mathbb{Z}$ and $\sim$ be $<$. In other words, $(n, m) \in \sim$ or $n \sim m$ exactly when $n<m$.
3. Set $A=\mathcal{P}(\mathbb{Z})$ which represents all the subsets of $\mathbb{Z}$ (called the power set of $\mathbb{Z})$. Let $\sim$ be $\subseteq$, i.e. subset. In other words, two subsets $X$ and $Y$ of $\mathbb{Z}$ will relate exactly when $X \subseteq Y$.
4. Take any set $A$ and let $\sim$ be equality, i.e. $a \sim b$ exactly when $a=b$. In other words $\sim={(a, a): a \in A}$.
5. Let $f: A \rightarrow B$ be a function from a set $A$ to another set $B$. Define a relation on $A$ as follows: $a \sim b$ iff $f(a)=f(b)$.

抽象代数代考

数学代写|线性代数代写线性代数代考|AXIOMATIC det

$$V^n=\underbrace{V \times V \times \cdots \times V}_{\mathrm{n} \text { times }}=\left{\left(v_1, v_2, \ldots, v_n\right): v_1, v_2, \ldots, v_n \in V\right}$$

1. $d\left(v_1, \ldots, v_i+v_i^{\prime}, \ldots, v_n\right)=d\left(v_1, \ldots, v_i, \ldots, v_n\right)+d\left(v_1, \ldots, v_i^{\prime}, \ldots, v_n\right)$和
2. $d\left(v_1, \ldots, a v_i, \ldots, v_n\right)=a d\left(v_1, \ldots, v_i, \ldots, v_n\right)$ for any $a \in \mathbb{R}$ .

Lemma $4.8$

1. $\left(v_1, v_2, \ldots, v_n\right)$ 包括一个坐标，它是零向量，那么 $d\left(v_1, v_2, \ldots, v_n\right)=0$n-线性函数的任何线性组合也是n-线性的。n-线性函数 $\mathbb{R}^n$ 是否完全由输入值
$$\left(e_{\sigma(1)}, e_{\sigma(2)}, \ldots, e_{\sigma(n)}\right),$$
where $e_1, e_2, \ldots, e_n$ 标准的依据是 $\mathbb{R}^n$ 和 $\sigma$ 这些数字有排列吗 $1,2, \ldots, n$.

数学代写|线性代数代写线性代数代考|QUOTIENT向量空间

1. 让 $A={a, b, c, d}$ 然后设置 $\sim={(a, b),(b, b),(c, d)}$。例如，根据我们的定义 $\sim$，我们有 $c \sim d$ 或 $c$ 涉及到 $d$.
2. 让 $A=\mathbb{Z}$ 和 $\sim$ 是 $<$。换句话说， $(n, m) \in \sim$ 或 $n \sim m$ 确切的时间 $n<m$.
3. 设置 $A=\mathcal{P}(\mathbb{Z})$ 的所有子集 $\mathbb{Z}$ 的幂集 $\mathbb{Z})$。让 $\sim$ 是 $\subseteq$，即子集。换句话说，两个子集 $X$ 和 $Y$ 的 $\mathbb{Z}$ 会准确地联系到 $X \subseteq Y$.
4. 取任意一组 $A$ 让 $\sim$ 平等，即。 $a \sim b$ 确切的时间 $a=b$。换句话说 $\sim={(a, a): a \in A}$.
5. 让 $f: A \rightarrow B$ 是集合中的一个函数 $A$ 到另一组 $B$。定义上的关系 $A$ 具体如下: $a \sim b$ iff $f(a)=f(b)$.

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