Prove polynomial transformation is linear

Question

Suppose a polynomial transformation:

How do I prove the "closed under addition" property of linearity? I am trying this:

I try to expand the equation on the left hand side, but I don't get anything resembling the right hand side. Thoughts?

By the way, is this how I would prove any polynomial transformation is linear - by picking two arbitrary polynomials with unknown coefficients and manipulating them?

Answer 1

Take $p,q$ two polynomials. Then $p+q$ is a polynomial defined as $(p+q)(x)=p(x)+q(x)$

$$ T(p+q)(x) = x^2(p+q)(x) = x^2 p(x) + x^2 q(x) = T(p)(x) + T(q)(x), $$

or,

$$ T(p+q) = T(p) + T(q). $$

In a similar fashion you have show the following; take $\alpha\in\mathbb{R}$:

$$ T(\alpha p)(x) = x^2\cdot (\alpha p)(x) = \alpha x^2 p(x) = \alpha T(p)(x), $$

or $T(\alpha p)=\alpha T(p)$. We conclude that $T$ is a linear operator.

Answer 2

Every polynomial of degree 2 is identified with an element of $\mathbb{R}^{3}$ namely (a.b.c). The same applies to polynomials of any degree. Every polynomial of degree n is identified with an element of $\mathbb{R}^{n+1}$. So your mapping is $T(a,b,c)=(a,b,c,0,0)$ where $T:\mathbb{R}^{3}\to \mathbb{R}^{5}$. Therefore your mapping is obviously linear because it represents a 5x3 matrix.!!