Pre-Quiz 2 Question
Could someone help me with number 4 parts b and c on the Pre-Quiz 2? The question was: Let V denote the set of all polynomials with the usual algebraic notions.
Part b: Show that the set of all polynomials f of degree less than 100 satisfying f(0) = f(1) = 0 is a subspace of V.
Part c: Show that the set of all polynomials f of degree less than 100 satisfying f(0) = f(1) = 1 is not a subspace of V.
I think what I am mostly struggling with is forming the correct set for f in both of the parts. Thank you so much!
Comments
By "forming the correct set", do you mean something like this description of the set of all polynomials of degree 100 or less:
$$\{a_{100} x^{100} + \cdots + a_1 x + a_0: a_{100},\ldots,a_1,a_0 \in \mathbb R\}?$$
If so, well, I wouldn't try! I just don't think you need a specific form for the polynomial to show, for example, that if $f(1)=g(1) = 0$, then $f(1)+g(1)=0$.
Thank you so much! This helped a bunch and really cleared things up.
How did you get started with this? I know that the two things that have to shown for it to be a subspace are closure under scalar multiplication and closure under vector addition, but I'm stuck on how to show those without having a specific form for the polynomial.
Edit: Never mind, I looked at some examples in the textbook (namely Example SP4 on page 211) and I figured it out.