GATE 2021 MA – Question 62
V is the real vector space of polynomials p=a₀+a₁x+a₂x². Define T(p)=[p(0)−p(1)]+[p(0)+p(1)]x+p(0)x². Find the sum of its eigenvalues.
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Correct answer: 1 to 1
Explanation
In basis (1,x,x²), the coefficient map is (a₀,a₁,a₂)↦(−a₁−a₂,2a₀+a₁+a₂,a₀). The matrix diagonal is (0,1,0), so its trace, equal to the eigenvalue sum counting multiplicity, is **1**.