Question 16 (CSIR NET June 2019)

Consider the vector space $P_n$ of real polynomials in $x$ of degree $\le n$. Define $T: P_2 \to P_3$ by $T(f(x)) = \int_0^x f(t) dt + f'(t)$. Find the matrix representation of $T$ with respect to the basis $\{1, x, x^2\}$ and $\{1, x, x^2, x^3\}$.

Explanation: Given that basis of $P_2 = \{1, x, x^2\}$, and basis of $P_3 = \{1, x, x^2, x^3\}$.

Now, $T(1) = \int_0^x 1 dt + 0 = (t)|_0^x = x = 0 \cdot 1 + 1 \cdot x + 0 \cdot x^2 + 0 \cdot x^3$.

$T(x) = \int_0^x x dt + 1 = \frac{x^2}{2} + 1 = 1 \cdot 1 + 0 \cdot x + \frac{1}{2} x^2 + 0 \cdot x^3$

$T(x^2) = \int_0^x x^2 dt + 2x = \frac{x^3}{3} + 2x = 0 \cdot 1 + 2 \cdot x + 0 \cdot x^2 + \frac{1}{3} x^3$

Thus we have, $[T]^B_{B'} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{3} \end{pmatrix}$, where $B$ is Basis of $P_2$ and $B'$ is Basis of $P_3$.