1. ## Linear space question

say you have the set of all functions defined at 1 with f(1)=0 or a homogeneous differential equation y''+ay'+by=0, these are both linear spaces because they satisfy closure under addition/multiplication but why isn't it closed if you change 0 to a number c?

2. ## Re: Linear space question

Originally Posted by mathnerd15
say you have the set of all functions defined at 1 with f(1)=0 or a homogeneous differential equation y''+ay'+by=0, these are both linear spaces because they satisfy closure under addition/multiplication but why isn't it closed if you change 0 to a number c?
suppose f(1)=c, and g(1)=c

(f+g)(1) = 2c

other than c=0, there is no number such that 2c=c so closure under addition is immediately violated

3. ## Re: Linear space question

I see the sum must be 0 but how exactly does this follow from addition closure which says that there is a unique sum x+y, isn't 2c unique? is there a deeper insight for differential equations and a relation to one to one injectiveness?

4. ## Re: Linear space question

Originally Posted by mathnerd15
I see the sum must be 0 since by definition (f+g)(1)=f(1)=g(1)=0 is there a deeper insight for differential equations and a relation to one to one injectiveness?
Yes. The differential equations of the form some left hand side = 0, are known as the homogeneous solutions to that differential equation. Because the right hand side is zero if you have two solutions to that equation then their sum is also a solution and in general the homogeneous solutions form a space and finding various bases for that space is important.

Some of the harder core math guys on here can expand on this I'm sure.

I remember that we'd use a series of solutions to the homogeneous equation to solve for boundary conditions using Fourier analysis.

5. ## Re: Linear space question

Originally Posted by mathnerd15
I see the sum must be 0 but how exactly does this follow from addition closure which says that there is a unique sum x+y, isn't 2c unique?
this isn't what closure says

closure says if you have a set of elements with an addition operation then if you add two elements of that set the sum is also in that set.

so if my set of functions if specified as

$$f_i(1)=c$$

then if closure is to be satisfied

$$(f_i+f_j)(1)=c\;\;\forall i,j$$

which is violated if $$c \neq 0$$

6. ## Re: Linear space question

Thanks very much!!! Apostol defines addition closure as- For every pair of elements x and y in V there corresponds a unique elemnt in V called the sum of x and y, denoted by x+y.
Yes that is so beautiful that the solutions to homogeneous equations form a space and superposition can be used. there is a really nice Braun Springer text on DE

7. ## Re: Linear space question

Originally Posted by mathnerd15
Thanks very much!!! Apostol defines addition closure as- For every pair of elements x and y in V there corresponds a unique elemnt in V called the sum of x and y, denoted by x+y.
Yes that is so beautiful that the solutions to homogeneous equations form a space and superposition can be used. there is a really nice Braun Springer text on DE
ok, this is still my form of closure as it specifies that (x+y) is in fact in V. It just goes one step further and says that the + operation is bijective.

8. ## Re: Linear space question

I wonder why Apostol adds bijection...there's a really nice online latex editor- I'm trying to get the type on this site to load with mathjax