Inner Product Space

To talk about “length” or “angle” in abstract spaces, we define the Inner Product .

Core Concepts

Inner Product: Generalization of the dot product $\langle u, v \rangle$ .
Norm (Length): Induced by the inner product $||\mathbf{v}|| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle}$ .
Orthogonality: Two vectors are orthogonal if their inner product is zero.

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Gram-Schmidt Orthonormalization

An algorithm to convert any basis into an Orthonormal Basis (orthogonal unit vectors). It works by sequentially subtracting the “projection” of the new vector onto the existing ones.

Orthogonal Transformations

Transformations that preserve the inner product (and thus lengths and angles).

Rotation Matrices are prime examples.