Linear Algebra. Proving differentiable functions are a vector space.

[datran]

Question: Show the set of all differentiable functions on (-infinity, +infinity) that satisfy f′ + 2f = 0 is a vector space.

I started the problem by assuming that f and g are both differentiable functions that satisfy this vector space.

Then I ran through the ten axioms of addition and scalar multiplication and proving that each one works.

I feel like that does not answer the question though since why would I need the equation f' + 2f = 0?

How does that equation come into play?

Thanks for any help provided.

 

[tiny-tim]

welcome to pf!

hi datran! welcome to pf!

I feel like that does not answer the question though since why would I need the equation f' + 2f = 0?

How does that equation come into play?

you have to prove eg that (f+g) satisfies that equation

(yes, i know it's obvious … but you still have to prove it!)

 

[datran]

Oh!

So I would do (f+g) = (f+g)' + 2(f+g) = 0

and same thing over and over for the 10 axioms.

So really f and g are like variables?

Thank you so much! That actually made many more problems clearer!

 

[Fredrik]

You can start by proving that the set of all differentiable functions from ℝ to ℝ with the standard definitions of addition and scalar multiplication is a vector space. (Looks like you've done that already). Denote this space by V. Define U={f in V|f'+2f=0}. U is by definition a subset of V. If you prove that U contains the 0 function and is closed under addition and scalar multiplication, you can conclude that U is a subspace of V.

 

[blue_raver22]

In order to prove that the set of all differentiable functions on (-infinity, +infinity) that satisfy f′ + 2f = 0 is a vector space, we need to show that it satisfies the ten axioms of a vector space: closure under addition, closure under scalar multiplication, commutativity of addition, associativity of addition, existence of an additive identity, existence of an additive inverse, distributivity of scalar multiplication over addition, distributivity of scalar multiplication over scalar addition, associativity of scalar multiplication, and existence of a multiplicative identity.

The equation f' + 2f = 0 is important because it is the defining property of the set of functions we are considering. It is the condition that all functions in this set must satisfy. This condition ensures that the set is closed under addition and scalar multiplication, as well as satisfying the other axioms.

To show closure under addition, we need to show that if f and g are both differentiable functions that satisfy f' + 2f = 0, then their sum f+g must also satisfy f' + 2(f+g) = 0. Similarly, to show closure under scalar multiplication, we need to show that if f is a differentiable function that satisfies f' + 2f = 0, then kf also satisfies f' + 2(kf) = 0 for any scalar k.

By using the properties of differentiation and the fact that f' + 2f = 0, we can easily show that the sum and scalar multiple of two functions that satisfy this equation will also satisfy it. This shows that the set is closed under addition and scalar multiplication.

The other axioms can also be proven using the equation f' + 2f = 0. For example, to show commutativity of addition, we can use the fact that differentiation is a linear operator and thus the order of addition does not matter. Similarly, the existence of an additive identity can be proven by showing that the function f(x) = 0 satisfies the equation f' + 2f = 0.

In conclusion, the equation f' + 2f = 0 is crucial in proving that the set of all differentiable functions on (-infinity, +infinity) that satisfy this equation is a vector space. It is the condition that ensures the set satisfies all ten axioms of a vector space.