1. ## Some Simple True/False

Hello everyone. I have a practice test and there are some T/F questions on it. I don't know the answers for sure but I have guesses. Can you please tell me what the answers are and possibly explain?

True/False?:
1. P2 is a Subspace of P3 (I think it's true)
2. D[0,1], the set of all differential functions over [0,1] is a subspace of C[0,1] (I think it's true)
3. {ax^2 + bx + c : b = 0} is a subspace of P2 (I think it's true....or maybe not)
4. There is a basis for P3 which contains the function f defined by f(x) = x^2 - x + 1 (True..(x^2, -x, 1)?)

2. Originally Posted by chickeneaterguy
Hello everyone. I have a practice test and there are some T/F questions on it. I don't know the answers for sure but I have guesses. Can you please tell me what the answers are and possibly explain?

True/False?:
1. P2 is a Subspace of P3 (I think it's true)
2. D[0,1], the set of all differential functions over [0,1] is a subspace of C[0,1] (I think it's true)
3. {ax^2 + bx + c : b = 0} is a subspace of P2 (I think it's true....or maybe not)
4. There is a basis for P3 which contains the function f defined by f(x) = x^2 - x + 1 (True..(x^2, -x, 1)?)

All of them are true, and here are some hints: (1) is any polynomial of degree at most 2 a pol. of degree at most 3? ; (b) Is every differentiable function a continuous one? ; (c) Is the set of all pol's of degree at most 2 with zero linear coefficient a vector space? ; (d) Is is true that any non-zero vector in any non-zero vector space is part of some basis?

Tonio

3. Originally Posted by tonio
All of them are true, and here are some hints: (1) is any polynomial of degree at most 2 a pol. of degree at most 3? ; (b) Is every differentiable function a continuous one? ; (c) Is the set of all pol's of degree at most 2 with zero linear coefficient a vector space? ; (d) Is is true that any non-zero vector in any non-zero vector space is part of some basis?

Tonio
Thanks.

(1) Makes sense... 0x^3 + x^2 + x + 1 = x^2 + x + 1
(2) Yeah. Makes sense. I was thinking C was something else for some reason.
(3) Yep. No explanation.
(4) I just learned basis...so I wasn't really sure.