Inner Products and Orthogonality

Define real (dot product) and complex (Hermitian) inner products and verify their properties.

Compute vector norms and distances derived from an inner product.

Determine orthogonality between vectors in both real and complex inner product spaces.

Apply the Gram-Schmidt process to construct an orthonormal basis from a given basis.

Compute the orthogonal projection of a vector onto a subspace.

Decompose a vector into components parallel and orthogonal to a subspace.

Define and identify orthogonal (real) and unitary (complex) matrices.

Describe how orthogonal and unitary operators preserve norms and inner products.