[Math] Kernel and Image of the differentiation transformation $T(p(x))=\frac{dp(x)}{dx}$

linear algebralinear-transformations

Let T:$P_3$$P_2$ be the differentiation transformation $T(p(x))=\frac{dp(x)}{dx}$.

Find bases for the kernel and the image space of $T$.

Best Answer

Let us see what happens to $p_3(x) = a_3x^3 + a_2 x^2 + a_1x + a_0$ after derivation. You get $p'(x)=3a_3 x^2 + 2a_2 x + a_1$. Namely, the original polynomial $p_3(x)$ is spanned by $\{x^3, x^2, x, 1\}$ where the Image is spanned by $\{x^2, x, 1\}$. As such, the polynomials which derivation is $0$ are constants, thus $KerT = sp\{1 \}$. You can easily check that $\dim P_3 = \dim P_2 + \dim P_0. $

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