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Questions › DSE › DSE 2020 › Q40
4 marks, −1Mathematics › Vector spaces, rank and eigenvalues › Norms, inner products and orthogonalityModerateabout 3 min

DSE 2020 Question 40: Vector spaces, rank and eigenvalues

Consider A=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)A=\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}A=(cosθsinθ​−sinθcosθ​) with θ∈[0,2π)\theta\in[0,2\pi)θ∈[0,2π), and the inner product ⟨v,w⟩=v1w1+v2w2\langle v,w\rangle=v_1w_1+v_2w_2⟨v,w⟩=v1​w1​+v2​w2​ on R2\mathbb{R}^2R2. For vectors v,w∈R2v,w\in\mathbb{R}^2v,w∈R2,

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Consider A=[cos θ -sin θ; sin θ cos θ] with θ∈[0,2π), and the inner product ⟨ v,w⟩=v₁w₁+v₂w₂ on ℝ². For vectors v,w∈ℝ², (A) ⟨ Av,Aw⟩=⟨ v,w⟩ (B) ⟨ Av,Aw⟩>⟨ v,w⟩ (C) ⟨ Av,Aw⟩<⟨ v,w⟩ (D) the comparison of ⟨ Av,Aw⟩ and ⟨ v,w⟩ depends on θ

Also written as: Consider A=[cos theta -sin theta; sin theta cos theta] with theta in [0,2pi), and the inner product < v,w>=v_1w_1+v_2w_2 on R^2. For vectors v,w in R^2,

Vector spaces, rank and eigenvalues has come up 41 times in 6 exams since 2008.

What this question tests

This question is on Norms, inner products and orthogonality, part of Vector spaces, rank and eigenvalues in Mathematics.

Length and angle in vector spaces and the matrices that preserve them.

TestStop has 7 past-paper questions on this subtopic; the nearest in year are below.

  • IIT JAM 2021 · Q10Let ‖·‖ and ⟨·,·⟩ denote the standard norm and inner product in ℝⁿ. If u, v ∈ ℝ³ with ‖u‖ = ‖v‖ = 2 and the…
Practise Vector spaces, rank and eigenvaluesMore on Norms, inner products and orthogonality

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