Orthogonality
Chapter 1: Vector Space and Inner Product — Orthogonality & Gram-Schmidt
From the book
Chapter 1: Vector Space and Inner Product. In the chapter mind map this icon labels Orthogonality & Gram-Schmidt. The discussion below is excerpted and lightly edited from § Gram–Schmidt Algorithm for Orthonormalization in Mathematics for AI and Machine Learning.
Let $W$ be a space of dimension $r$, and let ${\mathbf{v}_0, \ldots, \mathbf{v}_{r-1}}$ be a basis of $W$. We construct an orthonormal basis ${\mathbf{e}_0, \ldots, \mathbf{e}_{r-1}}$ as follows.
What this drawing shows
What you see. Shows perpendicular vectors or axes, representing zero inner product and independent geometric directions.
In the mind map. Chapter 1 — Orthogonality & Gram-Schmidt. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 1: Vector Space and Inner Product — see the mind-map node Orthogonality & Gram-Schmidt.