Show that $W$ is a subspace of $P_5$

Question

Let $W = \{ p(x) \in P_5 \mid p(0) = p(1) = 0 \}$.

a) Show that $W$ is a subspace of $P_5$.

b) Find a basis for $W$.

c) What is $\dim(W)$?

For part a), I need to show three things (I believe):

1) Closure under addition

2) Closure under multiplication

3) Show the vector zero is in the subset

I'm just unsure of how to show this closure with this example, I've done closure with other examples. I just don't know what $W$ really is, and what to do with it. Thanks for any help!

Answer 1

For closure note that $$ (P+Q)(0)=P(0)+Q(0) = 0+0 =0$$

$$ (\lambda P)(0)= \lambda (P(0))= \lambda 0 =0 $$

For basis consider $$\{ x(x-1), x^2(x-1), x^3(x-1) , x^4(x-1)\}$$

Thus the dimension of $W$ is four.

Answer 2

We are dealing with polynomials of degree 5 that is

$$p(x)=ax^5+bx^4+cx^3+dx^2+ex+f$$

with the following properties

• $p(0)=0 $
• $p(1)=0 $

Firstly let check that W is a subspace indeed

1. $p,q\in W\quad r=p+q\implies r(0)=p(0)+q(0)=0 \land r(1)=p(1)+q(1)=0\\\implies r\in W$
2. $p\in W\quad c\in\mathbb{R} \quad r(x)=cp(x) \implies r(0)=cp(0)=0 \land r(1)=cp(1)=0\implies r(x)\in W$
3. $r(x)=0 \implies r(0)=0 \land r(1)=0 \implies r(x)\in W$

To find a basis let find the explicit expression for $p(x)$ by the given properties

• $p(0)=0 \implies f=0$
• $p(1)=0 \implies a+b+c+d+e=0\implies e=-a-b-c-d$

that is

$$p(x)=ax^5+bx^4+cx^3+dx^2-(a+b+c+d)x=\\=a(x^5-x)+b(x^4-x)+c(x^3-x)+d(x^2-x)$$

thus a basis is

$$\{ (x^2-x), (x^3-x), (x^4-x) , (x^5-x)\}$$

from which we also deduce that the dimension of $W$ is $4$.

Answer 3

If you have a function T: $P_5$ -> R$^2$ where T(p) = [p(0) , p(1)], then W = {p:T(p)=0}. Given a vector space V, sets of the form {v:T(v)=0} are subspaces of V iff T is linear. So showing that W is subspace is equivalent to showing that T(ap+bq) = aT(p)+bT(q).

In other words, W is a subspace of V iff it there exists some linear operator for which W is the null space. So part (b) comes down to finding a basis of the null space of T, and (c) follows simply by counting the number of vectors in (b). It can also be found by finding the rank of T and subtracting it from the dimension of P$_5$.