### 数学代写|线性代数代写linear algebra代考|MATH1012

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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|Linear Combinations

Suppose we are working within a subspace for which all the radiographs satisfy a particular (significant) property, call it property $S$. This means that the subspace is defined as the set of all radiographs with property $S$. Because the subspace is not trivial (that is, it contains more than just the zero radiograph) it consists of an infinite number of radiographs. Suppose also that we have a handful of radiographs

that we know are in this subspace, but then a colleague brings us a new radiograph, $r$, one with which we have no experience and the colleague needs to know whether $r$ has property $S$. Since the set of radiographs defined by property $S$ is a subspace, we can perform a quick check to see if $r$ can be formed from those radiographs with which we are familiar, using arithmetic operations. If we find the answer to this question is “yes,” then we know $r$ has property $S .$ We know this because subspaces are closed under scalar multiplication and vector addition. If we find the answer to be “no, we still have more work to do. We cannot yet conclude whether or not $r$ has property $S$ because there may be radiographs with property $S$ that are still unknown to us.

We have also been exploring one-dimensional heat states on a finite interval. We have seen that the subset of heat states with fixed (zero) endpoint temperature differential is a subspace of the vector space of heat states. The collection of vectors in this subspace is relatively easy to identify: finitevalued and zero at the ends. However, if a particular heat state on a rod could cause issues with future functioning of a diffusion welder, an engineer might be interested in whether the subspace of possible heat states might contain this detrimental heat state. We may wish to determine if one such heat state is an arithmetic combination of several others.

In Section 3.1.1, we introduce the terminology of linear combinations for describing when a vector can be formed from a finite number of arithmetic operations on a specified set of vectors. In Sections $3.1 .3$ and 3.1.4 we consider linear combinations of vectors in Euclidean space ( $\left.\mathbb{R}^{n}\right)$ and connect such linear combinations to the inhomogeneous and homogeneous matrix equations $A x=b$ and $A x=0$, respectively. Finally, in Section 3.1.5, we discuss the connection between inhomogeneous and homogeneous systems.

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

We now assign terminology to describe vectors that have been created from (a finite number of) arithmetic operations with a specified set of vectors.

Let $(V,+, \cdot)$ be a vector space over $\mathbb{F}$. Given a finite set of vectors $v_{1}, v_{2}, \cdots, v_{k} \in V$, we say that the vector $w \in V$ is a linear combination of $v_{1}, v_{2}, \cdots, v_{k}$ if $w=a_{1} v_{1}+a_{2} v_{2}+\cdots+a_{k} v_{k}$ for some scalar coefficients $a_{1}, a_{2}, \cdots, a_{k} \in \mathbb{F}$.

Corollary 2.5.4 says that a subspace is a nonempty subset of a vector space that is closed under scalar multiplication and vector addition. Using this new terminology, we can say that a subspace is closed under linear combinations.
Following is an example in the vector space $\mathcal{I}_{4 \times 4}$ of $4 \times 4$ images.
Example 3.1.2 Consider the $4 \times 4$ grayscale images from page 11. Image 2 is a linear combination of Images $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ with scalar coefficients $\frac{1}{2}, 0$, and 1 , respectively, because

** Watch Your Language! When communicating whether or not a vector can be written as a linear combination of other vectors, you should recognize that the term “linear combination” is a property applied to vectors, not sets. So, we make statements such as
$\checkmark w$ is a linear combination of $v_{1}, v_{2}, v_{3}$
$\checkmark w$ is not a linear combination of $u_{1}, u_{2}, u_{3}, \ldots u_{n} \cdot$
$\checkmark w$ is a linear combination of vectors in $U=\left{v_{1}, v_{2}, v_{3}\right}$.
We do not say
$X w$ is a linear combination of $U$.
In the remainder of this section, we focus on the question: Can a given vector be written as a linear combination of vectors from some specified set of vectors?

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

In this section, we state definitions and give examples of matrix products. We expect that most linear algebra students have already learned to multiply matrices, but realize that some students may need a reminder. As with $\mathbb{R}, \mathcal{M}_{m \times n}$ can be equipped with other operations (besides addition and scalar multiplication) with which it is associated, namely matrix multiplication. But unlike matrix addition the matrix product is not defined component-wise.

Given an $n \times m$ matrix $A=\left(a_{i, j}\right)$ and an $m \times \ell$ matrix $B=\left(b_{i, j}\right)$, we define the matrix product of $A$ and $B$ to be the $n \times \ell$ matrix $A B=\left(c_{i, j}\right)$ where
$$c_{i, j}=\sum_{k=1}^{m} a_{i, k} b_{k, j}$$

We call this operation matrix multiplication.
Notation. There is no “.” between the matrices, rather they are written in juxtaposition to show a difference between the notation of a scalar product and the notation of a matrix product. The definition requires that the number of columns of $A$ is the same as the number of rows of $B$ for the product $A B$.
Example 3.1.8 Let
$$P=\left(\begin{array}{ll} 1 & 2 \ 3 & 4 \ 5 & 6 \end{array}\right), \quad Q=\left(\begin{array}{rr} 2 & 1 \ 1 & -1 \ 2 & 1 \end{array}\right), \text { and } R=\left(\begin{array}{rr} 2 & 0 \ 1 & -2 \end{array}\right)$$
Since both $P$ and $Q$ are $3 \times 2$ matrices, we see that the number of columns of $P$ is not the same as the number of rows of $Q$. Thus, $P Q$ is not defined. But, since $P$ has 2 columns and $R$ has 2 rows, $P R$ is defined. Let’s compute the matrix product $P R$. We can compute each entry as in Definition 3.1.7.
Position Computation
$$\begin{array}{ll} (i, j) & p_{i, 1} r_{1, j}+p_{i, 2} r_{2, j} \ \hline(1,1) & 1 \cdot 2+2 \cdot 1 \ (1,2) & 1 \cdot 0+2 \cdot(-2) \ (2,1) & 3 \cdot 2+4 \cdot 1 \ (2,2) & 3 \cdot 0+4 \cdot(-2) \ (3,1) & 5 \cdot 2+6 \cdot 1 \ (3,2) & 5 \cdot 0+6 \cdot(-2) \end{array}$$
Typically, when writing this out, we write it as
\begin{aligned} P R &=\left(\begin{array}{ll} 1 & 2 \ 3 & 4 \ 5 & 6 \end{array}\right)\left(\begin{array}{rr} 2 & 0 \ 1 & -2 \end{array}\right) \ &=\left(\begin{array}{rr} 1 \cdot 2+2 \cdot 1 & 1 \cdot 0+2 \cdot(-2) \ 3 \cdot 2+4 \cdot 1 & 3 \cdot 0+4 \cdot(-2) \ 5 \cdot 2+6 \cdot 1 & 5 \cdot 0+6 \cdot(-2) \end{array}\right) \ &=\left(\begin{array}{rr} 4 & -4 \ 10 & -8 \ 16 & -12 \end{array}\right) . \end{aligned}
In the above example, the result of the matrix product was a matrix of the same size as $P$. Let’s do another example to show that this is not always the case.

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

** 注意你说的话！在交流一个向量是否可以写成其他向量的线性组合时，您应该认识到术语“线性组合”是应用于向量而不是集合的属性。所以，我们做这样的陈述
✓在是一个线性组合在1,在2,在3
✓在不是的线性组合在1,在2,在3,…在n⋅
✓在是向量的线性组合U=\left{v_{1}, v_{2}, v_{3}\right}U=\left{v_{1}, v_{2}, v_{3}\right}.

X在是一个线性组合在.

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

C一世,j=∑ķ=1米一个一世,ķbķ,j

\begin{array}{ll} (i, j) & p_{i, 1} r_{1, j}+p_{i, 2} r_{2, j} \ \hline(1,1) & 1 \ cdot 2+2 \cdot 1 \ (1,2) & 1 \cdot 0+2 \cdot(-2) \ (2,1) & 3 \cdot 2+4 \cdot 1 \ (2,2) & 3 \cdot 0+4 \cdot(-2) \ (3,1) & 5 \cdot 2+6 \cdot 1 \ (3,2) & 5 \cdot 0+6 \cdot(-2) \end{数组}\begin{array}{ll} (i, j) & p_{i, 1} r_{1, j}+p_{i, 2} r_{2, j} \ \hline(1,1) & 1 \ cdot 2+2 \cdot 1 \ (1,2) & 1 \cdot 0+2 \cdot(-2) \ (2,1) & 3 \cdot 2+4 \cdot 1 \ (2,2) & 3 \cdot 0+4 \cdot(-2) \ (3,1) & 5 \cdot 2+6 \cdot 1 \ (3,2) & 5 \cdot 0+6 \cdot(-2) \end{数组}

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## MATLAB代写

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