这是一份nottingham诺丁汉大学ECON1043作业代写的成功案例

$\mathbf{x} \geq \mathbf{y} \Leftrightarrow x_{i} \geq y_{i} \quad$ for all $i=1,2, \ldots, n$
Readers of the literature need to be alert to what various authors mean by $\mathbf{x}>\mathbf{y}$ in $\mathfrak{R}^{n}$. We adopt the convention that
$\mathbf{x}>\mathbf{y} \Leftrightarrow x_{i}>y_{i} \quad$ for all $i=1,2, \ldots, n$
using $\mathbf{x} \gtreqless \mathbf{y}$ for the possibility the $\mathbf{x}$ and $\mathbf{y}$ are equal in some components $\mathbf{x} \gtreqless \mathbf{y} \Leftrightarrow x_{i} \geq y_{i} \quad$ for all $i=1,2, \ldots, n$ and $\mathbf{x} \neq \mathbf{y}$

Some authors use $>$ where we use $\gtreqless$, and use $\gg$ in place of $>$. Other conventions are also found.

The natural order is not the only way in which to order the product of weakly ordered sets. An example of a complete order on a product space $X=X_{1} \times X_{2} \times \cdots \times X_{n}$ is the lexicographic order, in which $\mathbf{x} \succ^{L} \mathbf{y}$ if $x_{k} \succ y_{k}$ in the first component in which they differ. That is,
$\mathbf{x} \succ^{L} \mathbf{y} \Leftrightarrow x_{k} \succ y_{k} \quad$ and $\quad x_{i}=y_{i} \quad$ for all $i=1,2, \ldots, k-1$

ECON1043 COURSE NOTES ：

Let $S$ be a nonempty subset of a linear space and let $m=\operatorname{dim}$ cone $S$. For every $\mathbf{x} \in$ cone $S$, there exist $\mathbf{x}{1}, \mathbf{x}{2}, \ldots, \mathbf{x}{n} \in S$ and $\alpha{1}, \alpha_{2}, \ldots, \alpha_{n} \in \mathfrak{R}{+}$ such that $$ \mathbf{x}=\alpha{1} \mathbf{x}{1}+\alpha{2} \mathbf{x}{2}+\cdots+\alpha{n} \mathbf{x}_{n}
$$

1. If $n>m=$ dim cone $S$, show that the elements $\mathbf{x}{1}, \mathbf{x}{2}, \ldots, \mathbf{x}{n} \in S$ are linearly dependent and therefore there exist numbers $\beta{1}, \beta_{2}, \ldots, \beta_{n}$, not all zero, such that
   $$
   \beta_{1} \mathbf{x}{1}+\beta{2} \mathbf{x}{2}+\cdots+\beta{n} \mathbf{x}_{n}=\mathbf{0}
   $$
2. Show that for any number $t, \mathbf{x}$ can be represented as
   $$
   \mathbf{x}=\sum_{i=1}^{n}\left(\alpha_{i}-t \beta_{i}\right) \mathbf{x}_{i}
   $$