Worked examples

Inner product, orthogonality and the spectral theorem

One sheet for each type of problem, with several worked examples of increasing difficulty: first the computation, then the explanation in words. Each sheet starts with links to the theory it uses.

• 1Norms, angles and orthogonality conditions between vectors
• 2Orthogonal projections onto a vector and onto a subspace
• 3Orthonormal bases with the Gram-Schmidt process
• 4Orthogonal complements of subspaces
• 5Orthogonal matrices and their completion
• 6Orthogonal diagonalizations with distinct eigenvalues
• 7Orthogonal diagonalizations with multiple eigenvalues