### 数学代写|数学分析代写Mathematical Analysis代考|The Dimension of a Vector Space

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## 数学代写|数学分析代写Mathematical Analysis代考|The Dimension of a Vector Space

In this section, we discuss the definition of dimension and prove the invariance of the cardinality of the basis. Some results on cardinal arithmetic are needed in the infinite-dimensional case. We also prove the existence of a vector space of any given dimension.

Definition. A vector space $U$ is said to be finite dimensional if it contains a finite basis.
Example 1. $\mathbb{K}^n$ and $\mathbb{P}n$ are finite dimensional. Lemma 3.3.1. Consider the following system of linear equations with coefficients in $\mathbb{K}$ : \begin{aligned} & a{11} x_1+a_{12} x_2+\ldots+a_{1 m} x_m=0 \ & a_{21} x_1+a_{22} x_2+\ldots+a_{2 m} x_m=0 \ & \vdots \ & \vdots \ & a_{n 1} x_1+a_{n 2} x_2+\ldots+a_{n m} x_n=0 . \end{aligned}
If $m>n$, then the system has a nontrivial (i.e., nonzero) solution $\left(x_1, \ldots, x_m\right) \in \mathbb{K}^m$.

Proof. Without loss of generality, assume that $m=n+1$, because we can augment the system by adding $m-n-1$ equations with zero coefficients to the system.
Since at least one of the coefficients is different from zero, we may assume, by reordering the equations and renumbering the variables, that $a_{11} \neq 0$. We prove the theorem by induction on $n$. Subtracting $\frac{a_{i, 1}}{a_{11}}$ times the top equation from equation $i, 2 \leq i \leq n$ yields the equivalent system
\begin{aligned} a_{11} x_1+a_{12} x_2+\ldots+a_{1, n+1} x_{n+1} & =0, \ b_{22} x_2+\ldots+b_{2, n+1} x_{n+1} & =0, \ \vdots & \vdots \ b_{n 2} x_2+\ldots+b_{n, n+1} x_{n+1} & =0, \end{aligned}
where $b_{i j}=a_{i j}-a_{i 1} a_{1 j} / a_{11}, 2 \leq i \leq n, 2 \leq j \leq n+1$. The bottom $n-1$ equations of the above system have a nontrivial solution $\left(x_2, \ldots, x_{n+1}\right)$, by the inductive hypothesis. Defining $x_1=\frac{-1}{a_{11}} \sum_{j=2}^{n+1} a_{1 j} x_j$ yields a nontrivial solution $\left(x_1, \ldots, x_{n+1}\right)$ of the original system.

## 数学代写|数学分析代写Mathematical Analysis代考|Linear Mappings, Quotient Spaces, and Direct Sums

A proper understanding of this section is essential for a smooth transition to the rest of the book. While the early results in the section are elementary, a number of important concepts make their first debut later in the section. Specifically, this includes quotient spaces and quotient maps, direct sums, projections and algebraic complements, linear functionals and linear operators, maximal subspaces and the co-dimension of a subspace and, finally, the definition of an algebra over a field.
Definition. Let $U$ and $V$ be vector spaces over $\mathbb{K}$. A mapping $T: U \rightarrow V$ is said to be linear if, for all $u, v \in U$, and all $a \in \mathbb{K}$,
$$T(u+v)=T(u)+T(v), \text { and } T(a u)=a T(u)$$
The following are examples of linear mappings.

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|The Dimension of a Vector Space

\begin{aligned} a_{11} x_1+a_{12} x_2+\ldots+a_{1, n+1} x_{n+1} & =0, \ b_{22} x_2+\ldots+b_{2, n+1} x_{n+1} & =0, \ \vdots & \vdots \ b_{n 2} x_2+\ldots+b_{n, n+1} x_{n+1} & =0, \end{aligned}

## 数学代写|数学分析代写Mathematical Analysis代考|Linear Mappings, Quotient Spaces, and Direct Sums

$$T(u+v)=T(u)+T(v), \text { and } T(a u)=a T(u)$$

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