### 数学代写|线性代数代写linear algebra代考|MATHS 1011

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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|Diffusion Welding and Heat States

In this section, we begin a deeper look into the mathematics for the diffusion welding application discussed in Chapter 1. Recall that diffusion welding can be used to adjoin several smaller rods into a single longer rod, leaving the final rod just after welding with varying temperature along the rod but with the ends having the same temperature. Recall that we measure the temperature along the rod and obtain a heat signature like the one seen in Figure $1.4$ of Chapter 1. Recall also, that the heat signature shows the temperature difference from the temperature at the ends of the rod. Thus, the initial signature (along with any subsequent signature) will show values of 0 at the ends.

The heat signature along the rod can be described by a function $f:[0, L] \rightarrow \mathbb{R}$, where $L$ is the length of the rod and $f(0)=f(L)=0$. The quantity $f(x)$ is the temperature difference on the rod at a position $x$ in the interval $[0, L]$. Because we are detecting and storing heat measurements along the rod, we are only able to collect finitely many such measurements. Thus, we discretize the heat signature $f$ by sampling at only $m$ locations along the bar. If we space the $m$ sampling locations equally, then for $\Delta x=\frac{L}{m+1}$, we can choose the sampling locations to be $\Delta x, 2 \Delta x, \ldots, m \Delta x$. Since the heat measurement is zero (and fixed) at the endpoints we do not need to sample there. The set of discrete heat measurements at a given time is called a heat state, as opposed to a heat signature, which, as discussed earlier, is defined at every point along the rod. We can record the heat state as the vector
$$u=\left[u_{0}, u_{1}, u_{2}, \ldots, u_{m}, u_{m+1}\right]=[0, f(\Delta x), f(2 \Delta x), \ldots, f(m \Delta x), 0]$$
Here, if $u_{j}=f(x)$ for some $x \in[0, L]$ then $u_{j+1}=f(x+\Delta x)$ and $u_{j-1}=f(x-\Delta x)$. Figure $2.15$ shows a (continuous) heat signature as a solid blue curve and the corresponding measured (discretized) heat state indicated by the regularly sampled points marked as circles.

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

We have seen that the set of discretized heat states of the preceding example forms a vector space. These discretized heat states can be viewed as real-valued functions on the set of $m+2$ points that are the sampling locations along the rod. In fact, function spaces such as $H_{m}(\mathbb{R})$ are very common and useful constructs for solving many physical problems. The following are some such function spaces.
Example 2.4.1 Let $\mathcal{F}={f: \mathbb{R} \rightarrow \mathbb{R}}$, the set of all functions whose domain is $\mathbb{R}$ and whose range consists of only real numbers.

We define addition and scalar multiplication (on functions) pointwise. That is, given two functions $f$ and $g$ and a real scalar $\alpha$, we define the sum $f+g$ by $(f+g)(x):=f(x)+g(x)$ and the scalar product $\alpha f$ by $(\alpha f)(x):=\alpha \cdot(f(x)) .(\mathcal{F},+, \cdot)$ is a vector space with scalars taken from $\mathbb{R}$.
Proof. Let $f, g, h \in \mathcal{F}$ and $\alpha, \beta \in \mathbb{R}$. We verify the 10 properties of Definition $2.3 .5$.

• Since $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ and based on the definition of addition, $f+g: \mathbb{R} \rightarrow \mathbb{R}$. So $f+g \in \mathcal{F}$ and $\mathcal{F}$ is closed over addition.
• Similarly, $\mathcal{F}$ is closed under scalar multiplication.
\begin{aligned} (f+g)(x) &=f(x)+g(x) \ &=g(x)+f(x) \ &=(g+f)(x) \end{aligned}
So, $f+g=g+f$
\begin{aligned} ((f+g)+h)(x) &=(f+g)(x)+h(x) \ &=(f(x)+g(x))+h(x) \ &=f(x)+(g(x)+h(x)) \ &=f(x)+(g+h)(x) \ &=(f+(g+h))(x) \end{aligned}
So $(f+g)+h=f+(g+h)$.
• We see, also, that scalar multiplication is associative. Indeed,
$$(\alpha \cdot(\beta \cdot f))(x)=(\alpha \cdot(\beta \cdot f(x)))=(\alpha \beta) f(x)=((\alpha \beta) \cdot f)(x)$$
So $\alpha \cdot(\beta \cdot f)=(\alpha \beta) \cdot f$

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

A matrix is an array of real numbers arranged in a rectangular grid, for example, let
$$A=\left(\begin{array}{lll} 1 & 2 & 3 \ 5 & 7 & 9 \end{array}\right)$$
The matrix $A$ has 2 rows (horizontal) and 3 columns (vertical), so we say it is a $2 \times 3$ matrix. In general, a matrix $B$ with $m$ rows and $n$ columns is called an $m \times n$ matrix. We say the dimensions of the matrix are $m$ and $n$.

Any two matrices with the same dimensions are added together by adding their entries entry-wise. A matrix is multiplied by a scalar by multiplying all of its entries by that scalar (that is, multiplication of a matrix by a scalar is also an entry-wise operation, as in Example 2.3.8).
Example 2.4.6 Let
$$A=\left(\begin{array}{lll} 1 & 2 & 3 \ 5 & 7 & 9 \end{array}\right), B=\left(\begin{array}{ccc} 1 & 0 & 1 \ -2 & 1 & 0 \end{array}\right), \text { and } C=\left(\begin{array}{ll} 1 & 2 \ 3 & 5 \end{array}\right)$$
Then
$$A+B=\left(\begin{array}{lll} 2 & 2 & 4 \ 3 & 8 & 9 \end{array}\right) \text {, }$$
but since $A \in \mathcal{M}{2 \times 3}$ and $C \in \mathcal{M}{2 \times 2}$, the definition of matrix addition does not work to compute $A+C$. That is, $A+C$ is undefined.
Using the definition of scalar multiplication, we get
$$3 \cdot A=\left(\begin{array}{lll} 3(1) & 3(2) & 3(3) \ 3(5) & 3(7) & 3(9) \end{array}\right)=\left(\begin{array}{ccc} 3 & 6 & 9 \ 15 & 21 & 27 \end{array}\right)$$
With this understanding of operations on matrices, we can now discuss $\left(\mathcal{M}_{m \times n},+, \cdot\right)$ as a vector space over $\mathbb{R}$.

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

• 自从F:R→R和G:R→R并基于加法的定义，F+G:R→R. 所以F+G∈F和F加法结束。
• 相似地，F在标量乘法下是闭合的。
• 加法是可交换的，因为
(F+G)(X)=F(X)+G(X) =G(X)+F(X) =(G+F)(X)
所以，F+G=G+F
• 并且，加法关联因为
((F+G)+H)(X)=(F+G)(X)+H(X) =(F(X)+G(X))+H(X) =F(X)+(G(X)+H(X)) =F(X)+(G+H)(X) =(F+(G+H))(X)
所以(F+G)+H=F+(G+H).
• 我们还看到，标量乘法是关联的。的确，
(一个⋅(b⋅F))(X)=(一个⋅(b⋅F(X)))=(一个b)F(X)=((一个b)⋅F)(X)
所以一个⋅(b⋅F)=(一个b)⋅F

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

3⋅一个=(3(1)3(2)3(3) 3(5)3(7)3(9))=(369 152127)

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## MATLAB代写

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