Linear transformation of polynomials, given transformation output

Question

I need to find a transformation of a polynomial, given the output of other polynomial calculations:

If $T : P_1 \mapsto P_2$ is a linear transformation such that, $$ T(1 + 5x) = 1 - 2x \quad\text{and}\quad T(3 + 14x) = -2 - 4x $$ Then $T(3 - 2x) =$ ?

Please explain this as simple as possible.

Answer 1

If you can write $$(3 - 2x) = a (1 + 5x) + b (3 + 14x)$$

then the linearity of $T$ implies that you can write

\begin{align*} T(3 - 2x) &= a T(1 + 5x) + b T(3 + 14x) \\ &= a(1 - 2x) + b(-2 - 4x) \\& = (a - 2b) + (-2a - 4b)x \end{align*}

So can you determine $a$ and $b$?

Answer 2

Write $3-2x$ in terms of $1+5x$ and $3+14x$. Specifically $3-2x = -52(1+5x) + 11(3+14x)$.

Now use linearity of $T$ to figure out what $T(-52(1+5x) + 11(3+14x))$ is.

Note that the $(-52,11)^T$ came from solving $\begin{bmatrix} 1 & 5 \\ 3 & 14 \end{bmatrix}^{-1} \begin{bmatrix} 3 \\ -2\end{bmatrix}$.

To elaborate: Since $3-2x = -52(1+5x) + 11(3+14x)$, we have \begin{eqnarray} T(3-2x) &=& T(-52(1+5x) + 11(3+14x)) \\ &=& -52 ( T(1+5x) ) + 11 ( T(3+14x) ) \\ &=& -52(1-2x) + 11(-2-4x) \\ &=& -74+60x \end{eqnarray}