数学代写|密码学代写cryptography theory代考|Essential Algebra

数学代写|密码学代写cryptography theory代考|Linear Algebra

While linear algebra has applications in many fields, including quantum physics and machine learning, it began as a way to solve systems of linear equations (thus the name). Linear equations are those for which all elements are of the first power. Thus, the following three equations are linear equations:
\begin{aligned} & a+b=10 \ & 3 x+5=44 \ & 4 x+2 y-z=35 \end{aligned}
However, the following are not linear equations:
\begin{aligned} & 2 x^2+2=10 \ & 4 y^2+4 x+3=17 \end{aligned}
The first three equations have all elements to the 1st power (often that with a number such as $x^1$ the 1 is simply assumed and not written). But in the second set of equations, at least one element is raised to some higher power. Thus, they are not linear. While linear algebra was created for the express purpose of solving linear equations, it has applications beyond that. In this chapter, the focus on linear algebra is in relation to certain cryptographic algorithms we will see in Chap. 20. These algorithms are lattice based. Understanding those algorithms requires at least a basic working knowledge of linear algebra.

When linear algebra is used, numbers are presented in the form of a matrix. And in this section, we will examine how to perform a variety of operations using matrices. Before we can delve into matrix math, we will need to define what a matrix is. A matrix is a rectangular arrangement of numbers in rows and columns. Rows run horizontally, and columns run vertically. The dimensions of a matrix are stated ” $m$

数学代写|密码学代写cryptography theory代考|Determinants

Next, we will turn our attention to another relatively easy computation, the determinant of a matrix (Wilkinson et al. 2013). The determinant of a matrix A is denoted by $|A|$. Calculating the determinant is not overly difficult. The question becomes: Why are we interested in this value? The determinant can be used for any number of things, but for our current purposes, it is necessary to calculate the determinant in order to find eigenvalues and eigenvectors. An example of a determinant in a generic form is shown below:
$$|A|\left[\begin{array}{ll} a & b \ C & d \end{array}\right]=a d-b c$$
A more concrete example might help elucidate this concept.
$$|A|\left[\begin{array}{ll} 2 & 3 \ 1 & 2 \end{array}\right]=(2)(2)-(3)(1)=1$$
A determinant is a value which is computed from the individual elements of a square matrix. It provides a single number, also known as a scaler value. Only a square matrix can have a determinant. The calculation for a $2 \times 2$ matrix is simple enough; we will explore more complex matrices in just a moment. However, what does this single scalar value mean? There are many things one can do with a determinant, most of which we won’t use in this text. It can be useful in solving linear equations and changing variables in integrals (yes linear algebra and calculus go hand in hand). But what is immediately useable for us is that if the determinant is non-zero, then the matrix is invertible. This will be important later.
What about a $3 \times 3$ matrix, such as Eq. 5.16

密码学代考

数学代写|密码学代写cryptography theory代考|Linear Algebra

$$a+b=10 \quad 3 x+5=444 x+2 y-z=35$$

$$2 x^2+2=10 \quad 4 y^2+4 x+3=17$$

数学代写|密码学代写cryptography theory代考|Determinants

$$|A|\left[\begin{array}{lll} a & b C & d \end{array}\right]=a d-b c$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: