Little bit confuse on vector space

In summary, the solution set of a second order differential equation, af''+bf'+cf=0, forms a real vector space with the usual operations of addition and scalar multiplication. This is because the set of differentiable functions is itself a vector space, and the solutions of the differential equation would form a subspace. To solve this type of equation, one can use the method of reducing it into a first order matrix system or by using exponential functions. Both methods will lead to a solution that is a linear combination of exponential functions.
  • #1
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how to proof if the solution set of a second order diffential equation af''+bf'+cf=0 is a real vector space w.r.t. the usual opeations?
 
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  • #2
Since the set of differentiable functions is itself a vector space the solutions would form a subspace. It thus is sufficient to show that the set is closed under the operations of addition and scalar multiplication.

Given any subset of a vector space you already have all the properties of associativity, distribution under scalar multiplication and vector addition, etc. The only issue is closure under the basic operations.
 
  • #3
james,

first of all thank you very much for your explaining, that helps me a lot.

but still i have some question to ask you, can you please help me out as well?

i just got no idea what is the set of the soluiton of those d.e.
do i need to use y=a^ex to solve them or i need to reduce them into first order matrix system?

but if i reduce into first order matrix system, how can i proof it is closeure under addition and scalar multiplication?

many thanks!

regards,
tony
 
  • #4
Either method works for finding solutions but solving the system directly is the most straightforward (presuming a,b, and c are constants). The way to look at this equation is in terms of the derivative as an operator D:

[aD^2 +bD + c1] f = 0
The exponential function f=e^(rx) is an "eigen-vector" of the D operator with eigen-value r. The set of all such function forms an "eigen-basis" so any solution must be a linear combination of exponential functions and you can find the r's algebraically.
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects called vectors and a set of operations that can be performed on these vectors, such as addition and scalar multiplication. In order for a set to be considered a vector space, it must satisfy certain axioms or properties, such as closure under addition and scalar multiplication, and the existence of an additive identity and inverse.

2. How is a vector space different from a regular space?

A vector space is a mathematical concept that is used to describe certain mathematical objects, while a regular space is a physical concept that describes our physical surroundings. In a vector space, the objects are abstract vectors that can have any number of dimensions, while in a regular space, the objects are physical entities that exist in three dimensions.

3. Can you give an example of a vector space?

Yes, an example of a vector space is the set of all real numbers, denoted by ℝ. In this vector space, the vectors are one-dimensional and can be represented as points on a number line. Addition and scalar multiplication can be performed on these vectors, and they satisfy all the axioms of a vector space.

4. How is a vector space useful in science?

Vector spaces are used in many scientific fields, such as physics, engineering, and computer science. They provide a mathematical framework for understanding and solving problems that involve quantities with both magnitude and direction, such as velocity, force, and electric fields. Vector spaces also allow for the manipulation and analysis of large sets of data, making them essential in data science and machine learning.

5. What are some applications of vector spaces?

Vector spaces have numerous applications in various fields, some of which include computer graphics, robotics, signal processing, and quantum mechanics. They are also used in linear algebra, which is the foundation of many mathematical and scientific fields. Additionally, vector spaces are used in optimization problems, where the goal is to find the best possible solution given a set of constraints.

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