Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Linear Algebra Linear Dependence, Basis & Dimension
Consider the vector space Pn of real polynomials in x of degree less than or equal to n. Define T : P2 → P3 by (Tf) (x) = \(\int_0^x f(t) d t+f^{\prime}(x)\) Then the matrix representation of T with respect to the bases {1, x, x2} and {1, x, x2, x3} is
1
\(\left(\begin{array}{cccc}0 & 1 & 0 & 0 \\ 1 & 0 & \frac{1}{2} & 0 \\ 0 & 2 & 0 & \frac{1}{3}\end{array}\right)\)
2
\(\left(\begin{array}{ccc}0 & 1 & 0 \\ 1 & 0 & 2 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{3}\end{array}\right)\)
3
\(\left(\begin{array}{cccc}0 & 1 & 0 & 0 \\ 1 & 0 & 2 & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{3}\end{array}\right)\)
4
\(\left(\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & \frac{1}{2} \\ 0 & 2 & 0 \\ 0 & 0 & \frac{1}{3}\end{array}\right)\)
5
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