Linear Algebra: UNIT III: Inner Product Spaces

The Gram Schmidt Orthogonalization: Theorem Part 2

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Important Theorems for Engineering Maths or Mathematics - The Gram Schmidt Orthogonalization: Theorems Part 2

THE GRAM SCHMIDT ORTHOGONALIZATION

THEOREM PART 2

Theorem 7

Let V be a non−zero finite dimensional inner product space. Then V has an orthonormal basis β. Further more if β= { v₁, v₂, ... v_n} and x ∈ V then x = ⁿΣ_i=1<x,v_i> v_i.

x =<x,vi> vi

Proof

Let β₀ be an ordered basis for V. We know that an orthogonal set β' of non zero vectors with span (β') = V. By normalizing each vector in β', we obtain an orthonormal set β span (β₀) that generates V. β is linearly independent. Β is an orthonormal basis for V.

Linear Algebra: UNIT III: Inner Product Spaces : Tag: maths, mathematics : - The Gram Schmidt Orthogonalization: Theorem Part 2

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Linear Algebra: UNIT III: Inner Product Spaces

Introduction of Inner Product Spaces

Inner Products and Norms: Definition, Theorem

Inner Products and Norms: Example Solved Problems - Part 1

Definitions of Unit vector, Orthogonal, Orthonormal

Inner Products and Norms: Example Solved Problems - Part 2

The Gram Schmidt Orthogonalization: Theorems Part 1

The Gram Schmidt Orthogonalization: Theorems Part 1 - Example Solved Problems

The Gram Schmidt Orthogonalization: Theorem Part 2

The Gram Schmidt Orthogonalization: Theorems Part 2 - Example Solved Problems

The Gram Schmidt Orthogonalization: Theorem Part 3

The Gram Schmidt Orthogonalization: Theorem Part 3 - Example Solved Problems

The Gram Schmidt Orthogonalization: Theorems Part 4

The Gram Schmidt Orthogonalization: Theorems Part 4 - Example Solved Problems

Adjoint of Linear Operator

Adjoint of Linear Operator: Example Solved Problems

Least Squares Approximation

Least Squares Approximation: Example Solved Problems

Inner Product Spaces: Exercise - Linear Algebra

Inner Product Spaces : 2 Marks Important Questions With Answer - Linear Algebra

Inner Product Spaces : Multiple Choice Questions and Answers - Linear Algebra

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