Inner Product Spaces and Orthogonality

GRE Math Subject · Axiom Academy

LESSON Inner Product Spaces and Orthogonality Angles, lengths, and orthogonal decompositions in abstract vector spaces An inner product on a vector space is a function satisfying: The norm induced by an inner product is: Vectors and are orthogonal if . Pythagorean theorem: If , then Advantages of orthonormal bases: Coordinates are easy to compute: Gram-Schmidt orthogonalization: Any basis can be converted to an orthonormal basis: The orthogonal projection of onto a subspace with orthonormal basis is: The vector is orthogonal to and gives the unique orthogonal decomposition: This is used to solve least-squares problems: minimize by solving . Compute norms and check orthogonality Apply Gram-Schmidt to find orthonormal bases Use orthonormal bases for easy coordinate computation Understand orthogonal projections and their properties Properties of orthogonal/unitary matrices (columns form orthonormal basis) Key fact: Orthogonal matrices satisfy , so . They preserve norms and angles!

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