GATE Linear Algebra: Vector Spaces and Linear Transformations – Previous Year Questions

11 GATE previous year questions on Vector Spaces and Linear Transformations (Linear Algebra, Mathematics) with answers and explanations, from every paper.

1. GATE 2021 MA Q11 (1 mark, Multiple choice) – A is real 3×4 and B real 4×3, with AB nonsingular. P:nullity A=0. Q:BA is nonsingular. Choose truth values.
2. GATE 2021 MA Q14 (1 mark, Multiple choice) – R is the reduced row echelon form of a real 4×4 A, and its third column is (0,1,0,0)ᵀ. P:if (α,β,γ,0)ᵀ solves Ax=0, then γ=0. Q:rank[A b]=rank[R b]…
3. GATE 2021 MA Q29 (1 mark, Numerical answer) – T:ℝ³→ℝ³ is linear and T(1,1,1)=(1,−1,1),T²(1,1,1)=(1,1,1),T²(1,1,2)=(1,1,1). Find rank T.
4. GATE 2022 MA Q12 (1 mark, Multiple choice) – P:if M∈ℝᵐˣⁿ,m>n≥2 and rank M=n, then Mx=0 has only x=0. Q:if nonzero E∈ℝⁿˣⁿ satisfies E³=0, then I+E² is singular. Choose truth values.
5. GATE 2022 MA Q47 (2 marks, Multiple choice) – A linear T:ℝ²→ℝ² satisfies T(1,2)=(1,0),T(2,1)=(1,1). If T⁻¹(p,q)=(x,y), which holds?
6. GATE 2022 MA Q56 (2 marks, Multiple select) – In ℝ³ set W₁=x:x₁+2x₂−x₃=0,W₂=x:2x₁+3x₃=0. Which hold?
7. GATE 2023 MA Q14 (1 mark, Multiple choice) – 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 holds?
8. GATE 2024 MA Q30 (1 mark, Numerical answer) – In ℝ⁴ with standard inner product, W₁=span(1,0,0,−1) and W₂=span(2,1,0,−1). Find dim(W₁⊥∩W₂⊥).
9. GATE 2024 MA Q60 (2 marks, Numerical answer) – H⊆S₃ consists of permutations σ for which trace(A₁A₂A₃)=trace(A σ(1)A σ(2)A σ(3)) for every complex 2×2 triple. Find H (integer).
10. GATE 2026 MA Q11 (1 mark, Multiple choice) – Let M₃(ℝ) be all real 3×3 matrices and T₃(ℝ) its upper-triangular subspace, with usual operations. To which listed real vector space is M₃(ℝ)/T₃(ℝ)…
11. GATE 2026 MA Q58 (2 marks, Numerical answer) – P₃(ℝ) contains real polynomials of degree at most three. T:P₃(ℝ)→ℝ² sends p to (p(1),p′(1)). Find nullity (integer).