CSIR UGC NET June 2019 Part B Question 27

Question

Consider the vector space $\mathbb{P}_{n}$ of real polynomials in $x$ of degree less than or equal to $n$. Define $T: \mathbb{P}_{2} \rightarrow \mathbb{P}_{3}$ by $(T f)(x)=\int_{0}^{x} f(t) d t+f^{\prime}(x).$ Then the matrix representation of $T$ with respect to the bases $\left\{1, x, x^{2}\right\}$ and $\left\{1, x, x^{2}, x^{3}\right\}$ is

1. $\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 1 & 0 & \frac{1}{2} & 0 \\ 0 & 2 & 0 & \frac{1}{3}\end{array}\right)$

2. $\cdot\left(\begin{array}{lll}0& 1 & 0 \\ 1 & 0 & 2 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{3}\end{array}\right)$

3. $\left(\begin{array}{llll}0 & 1 & 0 & 0 \\ 1 & 0 & 2 & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{3}\end{array}\right)$

4. $\left(\begin{array}{ccc}0 & 1 & 0 \\ 1 & 0 & \frac{1}{2} \\ 0 & 2 & 0 \\ 0 & 0 & \frac{1}{3}\end{array}\right)$