# Prove this is a Real Vector Space

• fk378
In summary, the homework statement is trying to find a function that satisfies the following:- d^2(y)/(dx^2) + 9y=0- yz dx is an integral- f(x) is a function of x- g and h are elements of V- a and b are real numbers- g + h is in V if and only if g and h are in V- to prove that V satisfies the axioms, the student needs to use the theorems of differentiability.
fk378

## Homework Statement

Let V be the real functions y=f(x) satisfying d^2(y)/(dx^2) + 9y=0.

a. Prove that V is a 2-dimensional real vector space.
b. In V define (y,z) = integral (from 0 to pi) yz dx. Find an orthonormal basis in V.

## The Attempt at a Solution

part A:
I integrated and got that f(x)= (-3/2)y^3 + Cy, C is a real number.
It seems like I need to use dot product here. I don't know how, though.

part B:
Completely lost.

First off, f(x) should be a function of x.
Second, you made a mistake in solving your DE: the solutions to y'' + 9y = 0 are not what you show.

What do you know about the characteristic equation of a linear DE?

Oh, I never took a DE class. All I did was integrate it twice, and I didn't even realize it wasn't a function of x. What should I do?

OK. You don't need to know how to solve the equation in order to verify that the set of all solutions to it is a vector space.
You know that you have to verify a set of axioms involving addition and scalar multiplication, right?

Yes..
Under addition we must show it is well-defined and
1. Associativity
2. commutativity
3. zero element
4. inverses
5. closed

Under multiplication we need to show well-definedness and
1. L and R distribution
2. associativity
3. id (unit)
4. zero element
5. closed

do I have it covered here?

Yes, those are the axioms that you have to verify.

Let f, g, and h be elements of V. (What does that imply in regard to your DE?. IOW, what exactly does it mean for a function to be an element of V?)
Let a and b be real numbers.

To pick a couple, #4 in the addition group and #4 in the scalar multiplication group:

If g is in V, is there another element h in V so that g + h = 0?
If g and h are in V, is g + h in V?

fk378 said:
Oh, I never took a DE class. All I did was integrate it twice, and I didn't even realize it wasn't a function of x. What should I do?
The point was that you wrote f(x)= (-3/2)y^3 + Cy. "y" is not "x"! If you had had d^2y/dx^2= a function of x, then you could integrate twice but you cannot integrate an expression in "y" with respect to x when you don't know y as a function of x.

Presumably you already know that addition of functions is associative and commutative, that f(x)= 0 for all x is the additive identity, etc and don't need to prove that here. All you really need to show is that if f(x) and g(x) are functions satisfying d^2f/dx^2+ 9f= 0 and d^2g/dx^2+ 9g= 0 then any linear combination of them, af(x)+ bg(x) for a and b any numbers, also satisfies that.

HallsofIvy said:
All you really need to show is that if f(x) and g(x) are functions satisfying d^2f/dx^2+ 9f= 0 and d^2g/dx^2+ 9g= 0 then any linear combination of them, af(x)+ bg(x) for a and b any numbers, also satisfies that.

Well how do I use the second derivative in helping me prove that V satisfies all the axioms?

By using the theorems of differentiability. If f and g are differentiable functions, then so is f + g, and (f+g)' = f' + g'. Also, if a is a constant, then af is differentiable, and (af)' = af'. These ideas can be extended to the next higher derivative.

## 1. What is a vector space?

A vector space is a set of elements, called vectors, that can be added together and multiplied by scalars (such as real numbers) to produce new vectors. This set of vectors must satisfy a set of axioms, or rules, in order to be considered a vector space.

## 2. What are the axioms that a vector space must satisfy?

A vector space must satisfy the axioms of closure, commutativity, associativity, existence of an additive identity, existence of additive inverses, existence of a multiplicative identity, and distributivity.

## 3. How do you prove that something is a real vector space?

In order to prove that something is a real vector space, you must show that it satisfies all of the axioms mentioned above. This can be done by providing specific examples and showing that they follow the axioms, or by using mathematical proofs.

## 4. Can a subset of a vector space also be a vector space?

Yes, a subset of a vector space can also be a vector space as long as it satisfies all of the axioms mentioned above. This is because the axioms do not depend on the size or dimension of the vector space, but rather the properties of the elements within the set.

## 5. What are some real-world examples of vector spaces?

Some real-world examples of vector spaces include the set of all 2D or 3D geometric vectors, the set of all polynomials of a certain degree, and the set of all continuous functions on a specific interval. These examples all satisfy the axioms of a vector space and can be added and multiplied by scalars to produce new vectors.

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