# Prove that W is a subspace of P_4(t)

## Homework Statement

Let W = {p(t) ∈ P4(t): p(0)=0 }. Prove that W is a subspace of P4(t)

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## The Attempt at a Solution

I know three things have to be true in order to be a subspace:
1. zero vector must exist as an element
2. if u and v are elements, u+v must be an element
3. if u is an element cu is an element

I'm a bit confused. I've done some searching on this forum + my textbook, and I haven't seen any questions in this particular format. Any help would be greatly appreciated.

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1. zero vector must exist

This is a bit weirdly phrased. The zero vector always exists in the larger space. The issue is that it must be an element of the subspace.

The zero vector is here the zero polynomial. Thus p(x)=0 for all x. Does that p lie in our subspace? What do we need to check for that?

I edited my original post for the first condition. I'm not sure how to check for p(x)=0 to lie in our subspace. I think I may be a bit confused by P_4(x). What does the P_4 mean?

I edited my original post for the first condition. I'm not sure how to check for p(x)=0 to lie in our subspace. I think I may be a bit confused by P_4(x). What does the P_4 mean?

Likely, it is the vector space of polynomial function of degree ≤4.

So examples of things in $P_4$ are

$p(x)=0,~p(x)=x+1,~p(x)=4x^4+2x+1$

An example of something not in there is

$p(x)=x^6+x^5+1$ (since the degree is 6 and thus not ≤4)

or

$p(x)=\sin(x)$ (since it is not a polynomial)

so P(x)=0 lies in our subspace of P_4 because it is an example element in P4

so P(x)=0 lies in our subspace of P_4 because it is an example element in P4

Not so quick!!

P(x) certainly lies in P4, no problem with that.
But now you have a subspace of P4.

That is, you only look at elements of P4 which satisfy p(0)=0.

So, for example

$$p(x)=x^2+x$$

lies in P4 AND it satisfies p(0)=02+0=0. So it is in our subspace.

But

$$p(x)=x^2+x+1$$

does lie in P4, but it has p(0)=1. So it is not in our subspace.

Alright! I get that. However, what I don't get is the question. I see you listed a few polynomials as examples that satisfy vector 0 being an element of the subspace, but what is the question asking for? There is no polynomial or equation associated with the question.

Ok, for anyone who has come across my problem via search, there are a few things that you need to know.

Span of vectors = subspace

p(t) = a0 + a1t1..+ a4t4

With p(0)=0 a0=0

W={a1t...a4t4
span = { t, t^2, t^3, t^4 }