suppressWarnings(
library(tinytex)
)For a function to be a linear transformation: \[ \begin{aligned} 1&. \: T(u_1 + u_2) = T(u_1) + T(u_2) \: \forall \: u_1, u_2 \: \epsilon \: U\\ 2&. \: T(\alpha u) = \alpha T(u) \: \forall \: u \: \epsilon \: U \end{aligned} \] Property 1: Let \(u_1 = a^\prime + b^\prime x + c^\prime x^2 and \: u_2 = d^\prime + e^\prime x + f^\prime x^2\) then: \[ \begin{aligned} T(u_1 + u_2) &= T((a^\prime + b^\prime x + c^\prime x^2) + (d^\prime + e^\prime x + f^\prime x^2)) \\ &= T((a^\prime + d^\prime) + (b^\prime + e^\prime)x + (c^\prime + f^\prime)x^2) \end{aligned} \] Observe that, \[ a = (a^\prime + d^\prime) \\ b = (b^\prime + e^\prime) \\ c = (c^\prime + f^\prime) \\ \] Hence, \[ \begin{aligned} T((a^\prime + d^\prime) + (b^\prime + e^\prime)x + (c^\prime + f^\prime)x^2) &= \begin{bmatrix} 2(a^\prime + d^\prime) - (b^\prime + e^\prime) \\ (b^\prime + e^\prime) + (c^\prime + f^\prime) \end{bmatrix} \\ &= \begin{bmatrix} (2a^\prime - b^\prime) + (2d^\prime - e^\prime) \\ (b^\prime + c^\prime) + (e^\prime + f^\prime) \end{bmatrix} \\ &= \begin{bmatrix} (2a^\prime - b^\prime) \\ (b^\prime + c^\prime) \end{bmatrix} + \begin{bmatrix} (2d^\prime - e^\prime) \\ (e^\prime + f^\prime) \end{bmatrix} \\ &= T_1 + T_2 \\ &= T(u_1) + T(u_2) \end{aligned} \] Property 2: \[ \begin{aligned} T(\alpha u) &= T(\alpha (a + bx + cx^2)) \\ &= T((\alpha a) + (\alpha b)x + (\alpha c)x^2) \\ &= \begin{bmatrix} 2(\alpha a) - (\alpha b) \\ (\alpha b) + (\alpha c) \end{bmatrix} \\ &= \begin{bmatrix} \alpha(2a - b) \\ \alpha(b + c) \end{bmatrix} \\ &= \alpha \begin{bmatrix} 2a - b \\ b + c \end{bmatrix} \\ &= \alpha T(u) \end{aligned} \] Both property 1 and 2 of a linear transformation has been met. Therefore \[T: P_2 \rightarrow C^2, T(a + bx + cx^2) = \begin{bmatrix}2a - b \\b + c \end{bmatrix}\] is a linear transformation.