3. Suppose that $T: P_4 \rightarrow P_4$ is the linear transformation defined by $T(p) = x^2 p''(x) + xp'(x) + p(x)$. (a) Show that $T$ is linear. (b) Consider the ordered bases $A = (1, x, \dots, x^4)$. Find the matrix $T_A,A$.
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Additivity: Let p and q be polynomials. We need to show that T(p + q) = Tp + Tq. T(p + q) = (x(p + q) + (p + q)x + (p + q)) = (xp + xq + px + qx + p + q) = (xp + px + p) + (xq + qx + q) = Tp + Tq Homogeneity: Let p be a polynomial and c be a scalar. Show more…
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