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Unformatted text preview: MAS 213: Linear Algebra II. Problem set #10. Tutorial on the 23rd of November. This week’s topics: • Inner product spaces: the axioms. • The norm in an inner product space. • The Cauchy-Schwarz inequality. • Orthogonality. • Orthonormal sets. Tutorial problems: Problem 1: (Problem 6.1 from [FIS].) Consider the inner product space C ([0 , 1]) equipped with the standard in- tegration inner product. Let f ∈ C ([0 , 1]) be defined by f ( t ) = t and let g ∈ C ([0 , 1]) be defined by g ( t ) = e t . (i) Calculate ⟨ f, g ⟩ , ∥ f ∥ , ∥ g ∥ , and ∥ f + g ∥ . (ii) Using these calculations, check an instance of the Cauchy-Schwarz in- equality, and check an instance of the triangle inequality. Problem 2: Let T be a linear operator on an inner product space V satisfying the property that ∥ T ( v ) ∥ = ∥ v ∥ for all v ∈ V . Prove that an operator with this property is 1-1. Comment: An operator with this property is called an isometry. The defining property of an isometry is that it preserves the length of a vector. Rotations and reﬂections are important examples of isometries. 1 Problem 3: In each of the following parts you are given a possible definition of an inner product space. In each part either prove that the given definition does indeed define an inner product space, or state (with justification) which axioms fail....
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- Fall '10