<s>[INST] Why isn’t $f:\!R \rightarrow P : a \rightarrow ax^3 + a^2x^5$ a linear operator? Why isn’t $f:\!R \rightarrow P : a \rightarrow ax^3 + a^2x^5$ a linear operator?
Because $0 = 0$ and in my understanding it is closed in linear combinations. Isn’t it?
 [/INST] Simply because $f(a+b) \neq f(a)+f(b)$.
$$f(a+b)=(a+b)x^3+(a+b)^2x^5=(a+b)x^3+(a^2+2ab+b^2)x^5$$
while
$$f(a)+f(b)=ax^3+a^2x^5+bx^3+b^2x^5=(a+b)x^3+(a^2+b^2)x^5$$
And they are different unless $ab=0$ i.e. $a=0$ or $b=0$
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