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Let $$T: P_2 \to P_2$$

Such that $$T(a_0 + a_1x + a_2x^2) = a_0 + a_1(x + 2) + a_2(x + 2)^2$$

Find the matrix $T$ relative to the standard basis $B$ for $P_2$: $$B = \{1, x, x^2\}$$

I understand the procedure for finding the matrix $T$, but how exactly do the vectors of $B$ plug into the equation? i.e $$T(1) = ?$$ $$T(x) = ?$$ $$T(x^2) = ?$$

Do we treat in the first example $a_0 = 1$ and in the second equation $a_1 = x$ and $a_3 = x^2$ in the last example?

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  • $\begingroup$ The $a$’s are the coefficients of powers of $x$ in the polynomial. $\endgroup$
    – amd
    Oct 19, 2016 at 7:32

1 Answer 1

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I imagine the result should be as follows, but I'm not sure?

$$T(1) = 0$$ $$T(x) = x + 2$$ $$T(x^2)= (x + 2)^2 = 4 + 4x + x^2$$

Which produces the matrix: $$ [T]_\beta \begin{bmatrix} 0 & 2 & 4 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \\ \end{bmatrix} $$

Am I on the right track?

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  • $\begingroup$ $T(1) = T(a_0) = a_0$, where $a_0 = 1$, so... $\endgroup$
    – Héctor Asencio L
    Oct 19, 2016 at 12:46
  • $\begingroup$ And $T(x) = T(a_1x) = x + 2$ where $a_1 = 1$? And $T(x^2) = T(a_2x^2) = 4 + 4x + x^2$ where $a_2 = 1$? $\endgroup$
    – Duane
    Oct 19, 2016 at 13:04

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