数学代写|Math5341 Linear algebra

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Math5341课程简介

This Module is offered by TUM Department of Mathematics.
This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Basic Information
Content, Learning Outcome and Preconditions
Content
In this course, we will cover basic techniques of structurepreserving or geometric numerical integration for ordinary differential equations.

1. Lagrangian and Hamiltonian Dynamics
2. Symplectic Integrators
3. Backward Error Analysis
4. Volume-Preserving Methods
5. Variational Integrators
6. Energy-Preserving Methods

Prerequisites 

Learning Outcome
After successful completion of the module, students are able to recognize various geometric structures present in many ordinary differential equations. They have an overview of state-of-theart numerical integration methods which preserve these structures and are able to select and implement suitable methods depending on the equations at hand and the desired conservation properties. Participating students are able to proof the conservation properties of the presented methods, either by direct computation, the discrete Noether theorem, or backward error analysis.
Preconditions
Linear Algebra and Discrete Structures 1 (MA1101)
Ordinary Differential Equations (MA2005)
Numerics of Ordinary Differential Equations (MA2304)

Math5341 Linear algebra HELP（EXAM HELP， ONLINE TUTOR）

Give complete answers to each of the following questions. Show all of your work.
Note: You might struggle with some of these questions, either because it has been a while since you took linear algebra, or because you did not learn some of these concepts. Don’t worry. We will use the Discussion Forum on https://space.uh.edu to ask questions and help one another.

Suppose $A$ is a real $m$ by $n$ matrix and $b \in R^m$.

Saying that a vector $x \in \mathbb{R}^n$ is a solution to the equation $A x=b$ means that when $x$ is substituted into the equation $A x$, the resulting vector is equal to $b$. Here, $A$ is an $m \times n$ matrix, $b$ is an $m$-dimensional column vector, and $x$ is an $n$-dimensional column vector. The equation $A x=b$ is a system of linear equations, and finding a solution to this system means finding a vector $x$ that satisfies all the equations in the system simultaneously.

In other words, $x$ is a solution to $A x=b$ if and only if $A x$ is equal to $b$, i.e., $A x=b$. If there is a unique solution to this system, then $x$ is called the solution to $A x=b$. If there are multiple solutions to the system, then $x$ is one of the solutions to $A x=b$. If there are no solutions to the system, then $x$ does not exist, and we say that the system is inconsistent.

Saying that the system $A x=b$ is consistent means that there exists at least one vector $x$ in $\mathbb{R}^n$ that satisfies all the equations in the system simultaneously. In other words, there is a solution $x$ to the system $A x=b$.

Geometrically, a consistent system represents a set of linear equations whose solution is a point or a collection of points in the $n$-dimensional space $\mathbb{R}^n$. The set of all solutions to a consistent system of linear equations is called the solution set of the system.

On the other hand, an inconsistent system is a set of equations that has no solution, meaning that there is no vector $x$ that satisfies all the equations in the system simultaneously. Geometrically, an inconsistent system represents a set of linear equations whose solution is an empty set or a set of parallel lines or planes that never intersect in $\mathbb{R}^n$.

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