Linear Algebra - subspaces of f in C[-1,1]

  • Thread starter Roni1985
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Homework Statement


Determine whether the following are subspaces of C[-1,1]:

d) The set of functions f in C[-1,1] such that f(-1)=0 AND f(1)=0


Homework Equations





The Attempt at a Solution



I did the question with 'OR', but I don't think I can find the functions.

I am not sure I can use x2 here ...

say f2(x)=f1(x)=x2-1 will give me 0 when x=-1 or x=1
 
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Answers and Replies

  • #2
Dick
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C[-1,1] doesn't have anything to with polynomials. You have to show if f(x) and g(x) satisfy your condition then so does f(x)+g(x) and c*f(x). Forget the polynomials.
 
  • #3
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C[-1,1] doesn't have anything to with polynomials. You have to show if f(x) and g(x) satisfy your condition then so does f(x)+g(x) and c*f(x). Forget the polynomials.
I just solved a similar question but with 'OR' instead of 'AND'.

I actually compared my answer with this file:
http://www.math.ohio-state.edu/~chou/Site/Homework_(571)_files/Practice_Final_solution.pdf [Broken]

they used the first polynomial to get the answer.

How would you suggest solving this one ?
I am kind of lost right here....
 
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  • #4
Dick
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You can use a polynomial to provide give a counterexample in the OR case. Because it's false and the polynomials are contained in C[-1,1]. You can't prove the case of AND just using polynomials because it is true. I already told you. Just show f(x) and g(x) vanish at both endpoints that if f(x)+g(x) and c*f(x) also have that property.
 
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  • #5
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You can use a polynomial to provide give a counterexample in the OR case. Because it's false and the polynomials are contained in C[-1,1]. You can't prove the case of AND just using polynomials because it is true.

Ohh, I see....

Just show f(x) and g(x) vanish at both endpoints that f(x)+g(x) and c*f(x) also have that property.

But, what are my f(x) and g(x) ?
If I can't use the polynomials, what can I use ?
are the simply the 0 functions ?
f(x)=0=g(x)
?
 
  • #6
Dick
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Ohh, I see....



But, what are my f(x) and g(x) ?
If I can't use the polynomials, what can I use ?
are the simply the 0 functions ?
f(x)=0=g(x)
?

f(x) and g(x) are just continuous functions on [-1,1]. All you know about them is that f(-1)=f(1)=0 and g(-1)=g(1)=0, since they are in the set that you are supposed to prove is a subspace. That's all you need to know. Isn't that enough?
 
  • #7
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f(x) and g(x) are just continuous functions on [-1,1]. All you know about them is that f(-1)=f(1)=0 and g(-1)=g(1)=0, since they are in the set that you are supposed to prove is a subspace. That's all you need to know. Isn't that enough?
OMG, you are so right ...

now it's crystal clear :\

thanks very much....
 

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