(Solved): Referring to problems 1) and 2): You are NOT allowed to use Matlab/Octave. Show all your work and b ...

Referring to problems 1) and 2): You are NOT allowed to use Matlab/Octave. Show all your work and box in all solutions. After the completion of the test, please submit the scan in a single PDF form. Suppose \( D: \mathbb{R}^{3}|x| \rightarrow \mathbb{R}^{2}|x| \) is linear and you know what it does to vectors in the basis \( B=\left\{1,1-x, x-x^{3}, x^{2}+2\right\} \) : \[ \begin{aligned} D(1) &=0 \\ D(1-x) &=1+x \\ D\left(x-x^{3}\right) &=1-3 x \\ D\left(x^{2}+2\right) &=x-x^{2} \end{aligned} \] a) Find \( D\left(2 x^{3}-3 x^{2}+x-4\right) \). (36 points) b) For given basis \( B^{\prime}=\left\{x, 1, x+x^{2}, x^{3}\right\} \) find \( M_{B}^{B^{\prime}}(i d) \). (36 points)