GATE 2020 MA – Question 59

Linear Algebra · Vector Spaces and Linear Transformations · 2 marks · Numerical answer

A linear map T:R⁴→R[x] satisfies $T(-1,1,1,1)=x^2+2x^4$, $T(1,2,3,4)=1-x^2$ and $T(2,-1,-1,0)=x^3-x^4$. Find the coefficient of x⁴ in T(−3,5,6,6).

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: 5 to 5

Explanation

The target vector is $2(-1,1,1,1)+(1,2,3,4)-(2,-1,-1,0)=(-3,5,6,6)$. By linearity its image is $2(x^2+2x^4)+(1-x^2)-(x^3-x^4)=1+x^2-x^3+5x^4$. The requested coefficient is 5.

More Vector Spaces and Linear Transformations questions

GATE 2026 MA Q11 – Let M₃(ℝ) be all real 3×3 matrices and T₃(ℝ) its upper-triangular subspace, with usual…
GATE 2026 MA Q58 – P₃(ℝ) contains real polynomials of degree at most three. T:P₃(ℝ)→ℝ² sends p to…
GATE 2024 MA Q30 – In ℝ⁴ with standard inner product, W₁=span(1,0,0,−1) and W₂=span(2,1,0,−1). Find…
GATE 2024 MA Q60 – H⊆S₃ consists of permutations σ for which trace(A₁A₂A₃)=trace(A σ(1)A σ(2)A σ(3)) for…
GATE 2023 MA Q14 – A linear T:ℝ³→ℝ³ satisfies T(1,0,0)=(0,1,1),T(1,1,0)=(1,0,1),T(1,1,1)=(1,1,2). Which…
GATE 2022 MA Q12 – P:if M∈ℝᵐˣⁿ,m>n≥2 and rank M=n, then Mx=0 has only x=0. Q:if nonzero E∈ℝⁿˣⁿ satisfies…

All Vector Spaces and Linear Transformations questions →