To prove the polynomial mapping is linear

Question

Show that the mapping $L : P_2 \to P_2$ given by $$L(p_0 + p_1x + p_2x^2 ) = p_1 + p_0x +(p_0 + 2p_1 − p_2)x^2$$ is linear and provide the matrix representing $L$ (with respect to the basis $\{1, x, x^2\}$).

Answer 1

Additivity:

$$ L(p_0 + p_1x + p_2x^2 ) = p_1 + p_0x + (p_0 + 2p_1 − p_2)x^2$$

$$ L(q_0 + q_1x + q_2x^2 ) = q_1 + q_0x + (q_0 + 2q_1 − q_2)x^2$$

Then $$ L((p_0+q_0) + (p_1+q_1) x + (p_2+q_2) x^2 ) = (p_1+q_1) + (p_0+q_0) x + ((p_0+q_0) + 2(p_1+q_1) − (p_2+q_2) )x^2=$$ $$ = p_1 + p_0x + (p_0 + 2p_1 − p_2)x^2+q_1 + q_0x + (q_0 + 2q_1 − q_2)x^2 =$$ $$ =L(p_0 + p_1x + p_2x^2 )+ L(q_0 + q_1x + q_2x^2 )$$

Homogeneity: $$ L(a(p_0 + p_1x + p_2x^2) ) = ap_1 + ap_0x + (ap_0 + a2p_1 − ap_2)x^2= aL(p_0 + p_1x + p_2x^2 )$$

Matrix:

$$L (1) = x+x^2$$ $$L(x) = 1+2x^2$$ $$L(x^2) = -x^2$$

$$L = \begin{pmatrix} 0 &1 &0 \\ 1& 0& 0\\ 1& 2& -1\\ \end{pmatrix}$$