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Let $V_4 = \{p(x) \in \mathbb{R}[X], \text{ such that } \deg{p(x)} \leq 4\}.$

Consider the linear map $f: V_4 \longrightarrow \mathbb{R}^2$ given by $$ p(x) \longmapsto \begin{bmatrix} p(1)\\\ p(2)\end{bmatrix}.$$

  1. Show that $\{(x-1)(x-2), x(x-1)(x-2), x^2(x-1)(x-2)\}$ is a basis of the nullspace $U$ of $f$.

  2. Show that $B = \{1, x, (x-1)(x-2), x(x-1)(x-2), x^2(x-1)(x-2)\}$ is a basis of $V_4$

  3. Compute the matrix of $f$ with respect to the basis $B$ of $V_4$ and $C =$ standard basis $\{(1,0), (0,1)\}$ of $\mathbb{R}^2$. I know the answer is: $$\begin{bmatrix} 1 & 1 & 0 & 0 & 0 \\\ 1 & 2 & 0 & 0 & 0 \end{bmatrix}$$ but I don't know how to reach this answer. It has something to do with computing $f(1), f(x)$ etc, which equal $(1,1)$ and $(1,2)$, respectively, but I don't understand how you work this out.

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    I tried to fix your formatting and typed it into LaTeX. Please check that I haven't introduced any mistakes.2011-04-16
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    thanks for that! it is dy not deg thats the only thing that is wrong! thanks. any clue as to the answers?2011-04-16
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    I don't understand the notation dy. I'm pretty sure it must be $\deg$ (the degree of a polynomial).2011-04-16

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