Linear space and Coordinates of vector

Question

Im taking a course on stepik.org on Linear Algebra. I come accross a problem...Please help....i mean how to initiate....

Thank you.

Answer 1

For linearity you have to consider to polynomials $p_1,p_2$ and $\lambda\in\mathbb K$ and prove $$ \varphi(p_1+p_2)=\varphi(p_1)+\varphi(p_2)\text{ and }\varphi(\lambda p_1)=\lambda\varphi(p_1) $$ Since you have to compare polynomials, you have to do it pointwise. So for an arbitrary $t$ you have to prove $$ \varphi(p_1+p_2)(t)=\varphi(p_1)(t)+\varphi(p_2)(t)\text{ and }\varphi(\lambda p_1)(t)=\lambda\varphi(p_1)(t) $$ Using the given notation, this is the same as $$ q_{p_1+p_2}(t)=q_{p_1}(t)+q_{p_2}(t)\text{ and }q_{\lambda p_1}(t)=\lambda q_{p_1}(t). $$ To find the matrix, you have to compute $\varphi(1)$, $\varphi(t)$ and $\varphi(t^2)$ and write it as a linear combination of the base $\{1,t,t^2,t^3\}$. The coefficients forms the matrix. More precisely: If $$ \varphi(1)=a_{11}1+a_{21}t+a_{31}t^2+a_{41}t^3\\ \varphi(t)=a_{12}1+a_{22}t+a_{32}t^2+a_{42}t^3\\ \varphi(t^2)=a_{13}1+a_{23}t+a_{33}t^2+a_{43}t^3\\ $$ then the matrix is $$ A=\begin{pmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ a_{41} & a_{42} & a_{43} \end{pmatrix} $$