Linear algebra basis/dimensions

[rosh300]

Homework Statement

Find the dimensions and basis of the following vector space V over the given field K:
a) V is the set of all polynomials over R (real) of degree at most n and whose coefficients add to 0, K = R (real numbers)
b) K = R (real), and V is the set of functions from R to R which are solutions of the differential equation: d²f/dx² - 9f = 0

Homework Equations

definition of basis, spanning and dimensions

The Attempt at a Solution

for part a) i think the dimetions is N and the basis is all the factors

for part b) i think it has 2 dimentions and the basis is df/dx - 3 and df/dx +3

 

[Dick]

Sort of. But what do you mean by 'factors' and df/dx-3 isn't a function from R->R. It's a differential equation. Can you spell out exactly what a basis is in each case?

 

[Mark44]

I might be wrong, but it appears that you "factored" this differential equation:

d²f/dx² - 9f = 0

to get this:

df/dx - 3 and df/dx +3

One problem with that is that the first DE had -9f, not -9.

 

[rosh300]

Sort of. But what do you mean by 'factors' and df/dx-3 isn't a function from R->R. It's a differential equation. Can you spell out exactly what a basis is in each case?

By basic i mean something which is linear independent and spans V

and by factors i mean suppose a₁, a₂, a₃ ... a_n were the roots, then the factors would be (x - a₁), (x - a₂), (x - a₃) ... (x - a_n) I suppose i should specfy the Real roots

 

[Dick]

By basic i mean something which is linear independent and spans V

and by factors i mean suppose a₁, a₂, a₃ ... a_n were the roots, then the factors would be (x - a₁), (x - a₂), (x - a₃) ... (x - a_n) I suppose i should specfy the Real roots

Exactly. In the first case you should specify N polynomials and in the second case you should specify two functions.