Essence of Linear Algebra (1): The Essence of Vectors — More Than Just Arrows

Wed, 01 Jan 2025 09:00:00 +0000

Why Vectors, and Why Care?#

A physicist talks about a force. A data scientist talks about a feature. A game programmer talks about a velocity. A quantum theorist talks about a state. Different fields, different terms — but the same underlying object: a vector.

That’s no coincidence. A vector is the simplest mathematical object flexible enough to describe anything you can add together and scale. Once you see this pattern, you’ll see it everywhere.

Functional Analysis (3): Hilbert Spaces — Geometry in Infinite Dimensions

Tue, 05 Oct 2021 09:00:00 +0000

Inner Products and the Geometry They Create#

If a Banach space is a normed space that has agreed to be complete, a Hilbert space is a Banach space that has further agreed to admit angles. That extra agreement — an inner product — is what restores almost all of finite-dimensional geometry to the infinite-dimensional setting. Orthogonality, projection, the Pythagorean theorem, the notion of “closest point in a subspace” — all come back unchanged. The price of admission is a single axiom; the reward, geometric and computational, is enormous.

Inner Product on Chen Kai Blog

Essence of Linear Algebra (1): The Essence of Vectors — More Than Just Arrows

Why Vectors, and Why Care?#

Functional Analysis (3): Hilbert Spaces — Geometry in Infinite Dimensions

Inner Products and the Geometry They Create#