Is the set of all continuous functions on the interval [0,1] a vector space?

[Saladsamurai]

Anyone else want to chime in? This is all new to me

You can't just say that a+b is polynomial of degree n with zero coefficients?

But like I said, this is new to me.

 

[Dick]

Anyone else want to chime in? This is all new to me

You can't just say that a+b is polynomial of degree n with zero coefficients?

But like I said, this is new to me.

nicksauce makes a good point. The question does say "degree EXACTLY n". The 'exactly' is likely there for a reason.

 

[Saladsamurai]

Okee-dokee. So since there is some polynomial of degree exactly n that when added to some other polynomial of exactly degree n does not YIELD a polynomial of exactly degree n, then the set of all polynomials of exactly degree n IS NOT a vector space.

So my coefficients of zero thing in post #36 is valid.

 

[Dick]

Right. You could also think of cases like p=x^2 and q=x^2-x, so p-q=x. Not a polynomial of degree EXACTLY two. If they had said polynomial of degree two or less, then it would be a vector space.