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Back to Orthogonality

Exercise 11.2.3

Avg difficulty 3.0

Consider the inner product space P3\mathbf{P}_{3} of polynomials of degree at most 33, with the inner product defined by

⟨f,g⟩=∫−11f(x)g(x) dx.\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)\,dx.

  1. Find the orthogonal complement of {x2,x}\left\{x^{2},x\right\}.

  2. Find the orthogonal complement of {x+1}\left\{x+1\right\}.

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Orthogonality, Exercise 11.2.3

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.