Linear algebra question?

One of my friends showed me how the following proof is done





Show that L: P_n -%26gt; P_n given by L(p) = p+p' is an isomorphism. First show that L is a linear operator. He said to do it this way:





Suppose p and q are in P_n. Then:


L(p+q) = p+q+(p+q)'


= p+p'+q+q'


= L(p)+L(q)


L(c*p)=c*p+(c*p)'


=c*p+c*p'


=c*(p+p')


=c*L(p)


So L is a linear operator.





Suppose that p is in the kernel of L. Then L(p) = 0. By definition of L, this means that p'=-p. The only element of P_n that would satisfy this equation is 0. This is because the derivative operator always lowers the degree of a polynomial by 1. Since the dimension of the kernel is 0 and P_n is finite dimensional, L is an isomorphism.





Now, my question is this:


How do we know that p'=-p? Is there a theorem stating this?

Linear algebra question?
It comes from the assumption that L(p) = 0. L(p) = p+p', so if p+p' = 0, then adding -p to both sides yields p'=-p.

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