Question

can someone help explain how to determine if something is a vector space or not

Answer 1

http://en.wikipedia.org/wiki/Vector_space#Definition

Answer 2

you have to check it for the axioms and see if they are true or not

Answer 3

how do you use the axioms to determine for instance if the set of all fifth degree polynomials with the standard operations is a vestor space

Answer 4

Is that space closed under addition

Answer 5

i guess there is a subtlety here. if it means the polynomial must have degree 5, then there is no zero vector. if it means degree 5 or less, then yes

Answer 6

right...as stated it is not a vector space...no zero vector and no closure of addition

Answer 7

that is the very thing that Im confused about how do you determine that there is no zero vector and no closure of addition

Answer 8

the zero vector has to be just the number 0

0 is not a 5th degree polynomial

Answer 9

\[(x^5+3x)+(-x^5+x^2)=3x-x^2\]

which is not a 5th degree polly

Answer 10

is there a vector (fifth degree polynomial in this case) say
\[p(x)\] with
\[p(x)+v(x)=v(x)\] for all fifth degree polynomials v
the answer is no, because the only polynomial that would work would be the zero polynomial, which does not have degree 5

Answer 11

further more,
\[x^5-2x+(-x^5)\] does not have degree 5, so it is not closed under addition

Answer 12

what zarkon said

Answer 13

ok that makes sense so what is the differance in saying it has fifth degree and fifth degree or less

Answer 14

is one a vector and one is not and why

Answer 15

5th degree or less is a vector space

Answer 16

so what is the step by step method to determing the answer to these types of problems. I mean when talking about continuous functions and things like that wouldnt get a little tricky

Answer 17

use the difinition I provided in my first post

Answer 18

i know it has to adhere to the list of axioms but can you explain it a little.