Question

Determine whether the following are subspaces of P4(Polynomial).


The set of all polynomial of degree 3

Answer 1

think you are missing the identity

Answer 2

?

Answer 3

ok sorry, what i guess i mean is you are missing the zero vector

Answer 4

one vector space axiom says there must be a 0, that is a vector
\[\overrightarrow{0}\] with
\[\overrightarrow{0}+\overrightarrow{v}=\overrightarrow{v}\]

Answer 5

if you have polynomials of degree three only, there is no zero vector

Answer 6

so in what cases would it have a zero vector?

Answer 7

you would need maybe all polynomials , so that would include the zero polynomial

Answer 8

or maybe all polynomials of degree 3 or less say, but it has to include zero

Answer 9

ok
thanks

Answer 10

but if you fix the degree it wont work
here is a simpler explanation, it has to be closed under vector addition so
\[x^3+(-x^3)=0\] tells you it is not even closed

Answer 11

so "the set of alll polynomial p(x) in P4 such that p(0)=0" would be subspace since it has zero vector?

Answer 12

i am not sure what P4 is, but i believe yes, that would be a subspace.
add them still works, subtract two
all the other axioms work
and it contains the zero polynomial for sure

Answer 13

you can check the axioms one by one (which is the point of this exercise) and see that they will all work

Answer 14

what do I add?

Answer 15

alpha
(a x^4+ b x^3 + c x^2 +d x )


(a x^4+ b x^3 + c x^2 +d x )+ (e y^4+ f y^3+g y^2+h y)=


?