# How to prove whether a set of polynomials is linearly independent?

$P_4(\mathbb{R})$ is the set of polynomials with degree at most $4$ with real coefficients.

$U=\{p \in P_4(\mathbb{R})∣p(3)=0\}$

A basis of $U$ would be $\{(x−3)^4,(x−3)^3,(x−3)^2,(x−3)\}$

But how do we prove, that the set of polynomial is linearly independent?

If we build the linear combination: $a\cdot(x−3)^4 + b\cdot(x−3)^3 + c\cdot(x−3)^2 + d\cdot(x−3) = 0$

Then we see that $a=b=c=d=0$. However, in order to be linearly independent, $a=b=c=d=0$ has to be the only solution for the equation. But this is not the case, since $x=3$ is also a solution $a\cdot(3−3)^4 + b\cdot(3−3)^3 + c\cdot(3−3)^2 + d\cdot(3−3) = 0$.

Could anyone tell me please, where is my mistake?

Thanks

• You can't really put $x=3$, you are taking linear combination of polynomials, you can't evaluate them at a certain point in this case. – QED Dec 4 '17 at 3:41

## 2 Answers

.Ah,no. It's a minor confusion, but worth clearing up all right.

See, as values, it is true that $$a(3-3)^4 + b(3-3)^3 + c(3-3)^2 + d(3-3) = 0$$. However, this does not mean that $$a(x-3)^4 + b(x-3)^3 + c(x-3)^2 +d(x-3)=0$$ as polynomials.

Equality as polynomials, means that the two polynomials must evaluate to the same quantity at every point. That is, $$p \equiv q$$ as polynomials if for all $$x$$, $$p(x)=q(x)$$.

You have only checked this for one point, $$x=3$$. There are many points left. As it turns out, $$a(x-3)^4 + b(x-3)^3 + c(x-3)^2 +d(x-3)=0$$ as polynomials, if and only if $$a=b=c=d=0$$. this can be proved by appropriate substitution of certain $$x$$, and then finding $$a,b,c,d$$ (or by differentiation).

I hope this clears the confusion.

• Oh, I see. Now I understand it. Thanks a lot :) – John Dec 4 '17 at 4:02
• You are welcome. Also, you may upvote answers if you find them satisfactory, including the one below. – Teresa Lisbon Dec 4 '17 at 4:02

The zero vector in your vector space is the zero polynomial, not the number $0$. Thus you need to find coefficients $a,b,c,d$ such that the polynomial evaluates to $0$ for all $x$, not just a particular $x$ such as your $x=3$. And only $a=b=c=d=0$ fits that bill.