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I am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial $ ax^2 + bx + c $ with the matrix $ A = \begin{bmatrix} 1,0,0 \\ 0,1,0 \\ 0,0,1 \\ \end{bmatrix} $ and the vector $ \begin{bmatrix} 1 \\ x \\ x^2 \\ \end{bmatrix} $ what corresponds to $a$, $b$, and $c$ in the matrix $A$?

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I think you need to be clear about what you mean by "representing" the polynomial. You can if you like make the assignments

$$ x^2 \;\; \to \;\; \left [ \begin{array}{c} 0\\ 0\\ 1\\ \end{array} \right ] \hspace{2pc} x \;\; \to \;\; \left [ \begin{array}{c} 0\\ 1\\ 0\\ \end{array} \right ] \hspace{2pc} 1 \;\; \to \;\; \left [ \begin{array}{c} 1\\ 0\\ 0\\ \end{array} \right ] $$

Then your polynomial can be represented by the vector

$$ ax^2 + bx + c \;\; \to\;\; \left [ \begin{array}{c} c\\ b\\ a\\ \end{array} \right ]. $$

To describe a linear transformation in terms of matrices it might be worth it to start with a mapping $T:P_2 \to P_2$ first and then find the matrix representation.

Edit: To answer the question you posted, I would take each basis vector listed above and apply the matrix to it:

\begin{eqnarray*} \left [ \begin{array}{ccc} 3 & 2 & 7 \\ 0 & 1 & 0 \\ 4 & 0 & 1 \\ \end{array} \right ] \left [ \begin{array}{c} 1 \\ 0\\ 0\\ \end{array} \right ] & = & \left [ \begin{array}{c} 3 \\ 0 \\ 4 \\ \end{array} \right ] \;\; \to \;\; 4x^2 + 3 \\ \left [ \begin{array}{ccc} 3 & 2 & 7 \\ 0 & 1 & 0 \\ 4 & 0 & 1 \\ \end{array} \right ] \left [ \begin{array}{c} 0 \\ 1\\ 0\\ \end{array} \right ] & = & \left [ \begin{array}{c} 2 \\ 1\\ 0\\ \end{array} \right ] \;\; \to \;\; x+ 2 \\ \left [ \begin{array}{ccc} 3 & 2 & 7 \\ 0 & 1 & 0 \\ 4 & 0 & 1 \\ \end{array} \right ] \left [ \begin{array}{c} 0 \\ 0\\ 1\\ \end{array} \right ] & = & \left [ \begin{array}{c} 7 \\ 0\\ 1\\ \end{array} \right ] \;\; \to \;\; x^2 + 7. \end{eqnarray*}

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  • $\begingroup$ I think I am looking at these spaces the wrong way. What does the matrix $ A = \begin{bmatrix} 3,2,7 \\ 0,1,0 \\ 4,0,1 \\ \end{bmatrix} $ correspond to, in terms of the basis $ A = \begin{bmatrix} 1 \\ x \\ x^2 \\ \end{bmatrix} $? $\endgroup$ Apr 23 '15 at 20:38
  • $\begingroup$ @120MinuteMan I edited my answer to answer this question. Let me know if it is clearer now. $\endgroup$
    – Mnifldz
    Apr 23 '15 at 20:45
  • $\begingroup$ Yes! Many thanks . . . it's starting to click now $\endgroup$ Apr 23 '15 at 20:48
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    $\begingroup$ What $\left [ \begin{array}{ccc} 3 & 2 & 7 \\ 0 & 1 & 0 \\ 4 & 0 & 1 \\ \end{array} \right ]$ mean? I know it's super late but I didn't get what that matrix mean actually. Is this a one kind of transformation on $P_2?$ @Mnifldz Sir $\endgroup$
    – emonHR
    Oct 23 '19 at 16:46
  • $\begingroup$ @emonhossain This was meant simply as an example. $\endgroup$
    – Mnifldz
    Oct 23 '19 at 18:11
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You shouldn't be representing the polynomial via the matrix $A$. Instead what you can think of is the following way of identifying a polynomial with a three dimensional vector. $$ax^2+bx+c \Leftrightarrow \begin{bmatrix}a\\b\\c\end{bmatrix}$$

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