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Questions › DSE › DSE 2018 › Q6
4 marks, −1Mathematics › Vector spaces, rank and eigenvalues › Norms, inner products and orthogonalityEasyabout 1 min

DSE 2018 Question 6: Vector spaces, rank and eigenvalues

Consider the matrix A=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}A=(cosθsinθ​−sinθcosθ​) where θ∈[0,2π)\theta \in [0, 2\pi)θ∈[0,2π). The inner product of vectors v=(v1,v2)v = (v_1, v_2)v=(v1​,v2​) and w=(w1,w2)w = (w_1, w_2)w=(w1​,w2​) in R2\mathbb{R}^2R2 is defined by ⟨v,w⟩≡v1w1+v2w2\langle v, w \rangle \equiv v_1 w_1 + v_2 w_2⟨v,w⟩≡v1​w1​+v2​w2​. So, for vectors vvv and www in R2\mathbb{R}^2R2,

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Consider the matrix A = [cos θ -sin θ; sin θ cos θ] where θ ∈ [0, 2π). The inner product of vectors v = (v₁, v₂) and w = (w₁, w₂) in ℝ² is defined by ⟨ v, w ⟩ ≡ v₁ w₁ + v₂ w₂. So, for vectors v and w in ℝ², (A) ⟨ Av, Aw ⟩ > ⟨ v, w ⟩ (B) The comparison of ⟨ Av, Aw ⟩ and ⟨ v, w ⟩ depends on the value of θ (C) ⟨ Av, Aw ⟩ = ⟨ v, w ⟩ (D) ⟨ Av, Aw ⟩ < ⟨ v, w ⟩

Also written as: Consider the matrix A = [cos theta -sin theta; sin theta cos theta] where theta in [0, 2pi). The inner product of vectors v = (v_1, v_2) and w = (w_1, w_2) in R^2 is defined by < v, w > == v_1 w_1 + v_2 w_2. So, for vectors v and 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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Question from the DSE 2018 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.