Linear algebra basis/dimensions

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Homework Statement


Find the dimensions and basis of the following vector space V over the given feild 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
 

Answers and Replies

  • #2
Dick
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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?
 
  • #3
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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.
 
  • #4
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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 ... an were the roots, then the factors would be (x - a1), (x - a2), (x - a3) ... (x - an) I suppose i should specfy the Real roots
 
  • #5
Dick
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By basic i mean something which is linear independent and spans V

and by factors i mean suppose a1, a2, a3 ... an were the roots, then the factors would be (x - a1), (x - a2), (x - a3) ... (x - an) 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.
 

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