# Linear Algebra homework problem

• Sampson12
In summary, the set of solutions to the differential equation f''(x)+3f'(x)+(x^2)f(x)=0 is a subspace of the vector space F.
Sampson12
The set F of all functions from R to R is a vector space with the usual operations of addition of functions and scalar multiplication. Is the set of solutions to the differential equation f''(x)+3f'(x)+(x^2)f(x)=0 a subspace of F? Justify your answer

I know that to prove that the set of solutions is a subspace of F I need to show that the set not empty, is closed under addition and closed under scalar multiplication. The only problem I have is solving the differential equation which i am not sure how to do because solving this kind of differential equation (I only know how to solve second order DE's with constant coefficients) has not brought up in the current course (Linear mathematics 2 year maths) or any of the prerequisite coursess . Do I actually need to solve the equation to find the answer or is their another way to find if its a subspace of F or not? Any help would be very much appreciated.

The problem does NOT ask you to solve the equation. Yes, you must show
1) the set is not empty. Equivalently show that it contains the "0" vector. Is f(x)= 0 a solution for this equation?

2) the set is closed under addition. If f and g are solutions, is f+ g as solution? Just put f+ g into the equation and try to separate f and g.

3) the set is closed under scalar multiplication. If f is a solution and a is a number, is af a solution. Just put af into the equation and try to factor out a.

if f(x) and g(x) satisfy your equation, you should be able to show that f(x)+g(x) also satisfies it. same goes for a*f(x). Finally you need to show that a solution exists; there exists theorems which state this, but it's easy to find a particular function which solves this equation (hint: existence of this solution follows directly from either of above conditions)

Thanks for the help

## 1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It involves the study of linear transformations, matrices, and systems of linear equations.

## 2. Why is Linear Algebra important?

Linear Algebra has many applications in fields such as physics, engineering, economics, and computer science. It is used to solve systems of linear equations, find patterns and relationships in data, and make predictions based on linear models.

## 3. What are the main topics covered in Linear Algebra homework?

The main topics covered in Linear Algebra homework include vector operations, matrices, systems of linear equations, determinants, eigenvalues and eigenvectors, and applications of these concepts.

## 4. How can I solve a Linear Algebra homework problem?

Solving a Linear Algebra homework problem involves understanding the concepts and applying the appropriate techniques to solve the given problem. It is important to read and understand the problem carefully, break it down into smaller steps, and use the relevant formulas and procedures to find the solution.

## 5. What are some tips for better understanding Linear Algebra concepts?

Some tips for better understanding Linear Algebra concepts include practicing regularly, seeking help from a tutor or teacher when needed, and breaking down complex problems into smaller steps. It is also helpful to visualize the concepts, use real-life examples, and relate them to other areas of mathematics.

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