### 数学代写|线性代数代写linear algebra代考|MTH 2106

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|线性代数代写linear algebra代考|Summations

I promised not to use equations full of Greek symbols in this book. However, there is one that is so common and useful that I would be remiss to not cover it. A summation is expressed as a sigma $\Sigma$ and adds elements together.

For example, if I want to iterate the numbers 1 through 5 , multiply each by 2 , and sum them, here is how I would express that using a summation. Example 1-10 shows how to execute this in Python.
$$\sum_{i=1}^{5} 2 i=(2) 1+(2) 2+(2) 3+(2) 4+(2) 5=30$$
Example 1-10. Performing a summation in Python
summation $=\operatorname{sum}(2 * i$ for i in range $(1,6))$
print(summation)
Note that $i$ is a placeholder variable representing each consecutive index value we are iterating in the loop, which we multiply by 2 and then sum all together. When you are iterating data, you may see variables like $x_{i}$ indicating an element in a collection at index $i$.

It is also common to see $n$ represent the number of items in a collection, like the number of records in a dataset. Here is one such example where we iterate a collection of numbers of size $n$, multiply each one by 10 , and sum them:
$$\sum_{i=1}^{n} 10 x_{i}$$
In Example 1-11 we use Python to execute this expression on a collection of four numbers. Note that in Python (and most programming languages in general) we typically reference items starting at index 0 , while in math we start at index 1 . Therefore, we shift accordingly in our iteration by starting at 0 in our range().

## 数学代写|线性代数代写linear algebra代考|Exponents

Exponents multiply a number by itself a specified number of times. When you raise 2 to the third power (expressed as $2^{3}$ using 3 as a superscript), that is multiplying three 2s together:
$$2^{3}=2 * 2 * 2=8$$
The base is the variable or value we are exponentiating, and the exponent is the number of times we multiply the base value. For the expression $2^{3}, 2$ is the base and 3 is the exponent.

Exponents have a few interesting properties. Say we multiplied $x^{2}$ and $x^{3}$ together. Observe what happens next when I expand the exponents with simple multiplication and then consolidate into a single exponent:
$$x^{2} x^{3}=\left(x^{} x\right)^{}\left(x^{} x^{} x\right)=x^{2+3}=x^{5}$$
When we multiply exponents together with the same base, we simply add the exponents, which is known as the product rule. Let me emphasize that the base of all multiplied exponents must be the same for the product rule to apply.
Let’s explore division next. What happens when we divide $x^{2}$ by $x^{5}$ ?
$$\frac{x^{2}}{x^{5}}$$ \begin{aligned} &\frac{x^{} x}{x^{} x^{} x^{} x^{} x} \ &\frac{1}{x^{} x^{*} x} \ &\frac{1}{x^{3}}=x^{-3} \end{aligned}
As you can see, when we divide $x^{2}$ by $x^{5}$ we can cancel out two $x^{\prime}$ ‘ in the numerator and denominator, leaving us with $\frac{1}{x^{3}}$. When a factor exists in both the numerator and denominator, we can cancel out that factor.

## 数学代写|线性代数代写linear algebra代考|Summations

$$\sum_{i=1}^{5} 2 i=(2) 1+(2) 2+(2) 3+(2) 4+(2) 5=30$$

summation中执行求和 $=\operatorname{sum}(2 * i$ 因为我在范围内 $(1,6))$

$$\sum_{i=1}^{n} 10 x_{i}$$

## 数学代写|线性代数代写linear algebra代考|Exponents

$$2^{3}=2 * 2 * 2=8$$

$$x^{2} x^{3}=(x x)(x x x)=x^{2+3}=x^{5}$$

$$\begin{gathered} \frac{x^{2}}{x^{5}} \ \frac{x x}{x x x x x} \quad \frac{1}{x x^{*} x} \frac{1}{x^{3}}=x^{-3} \end{gathered}$$

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## MATLAB代写

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