Overview of Inner Product Spaces, Orthogonality, Gram Schmidt Method, and Hilbert Spaces

In this video, we explore some of the foundational concepts in linear algebra that play crucial roles in quantum mechanics and advanced mathematics: inner product spaces, orthogonality, the Gram-Schmidt process, and Hilbert spaces.
We start by explaining the concept of inner products—how they help define angles, lengths, and the idea of orthogonality. We’ll see how the inner product provides us with a way to measure distances and determine when vectors are perpendicular, which leads us into the concept of orthogonality.
We look at why orthogonal vectors are so useful, showing how they represent independent directions and make calculations simpler. With a clear example involving force vectors, we’ll explore how orthogonal directions allow us to break complex problems down into manageable parts.
We then cover the Gram-Schmidt process—a method for creating an orthogonal basis from a set of non-orthogonal vectors. This method is essential in applications like physics and engineering. We illustrate this with a hands-on example that makes the process easy to follow and visually intuitive.
Finally, we introduce the concept of Hilbert spaces, an extension of vector spaces to infinite dimensions, which is a key framework for quantum mechanics. We explore how Hilbert spaces enable us to represent quantum states and perform calculations involving superposition and probabilities. You’ll see why quantum mechanics relies so heavily on the mathematical properties of Hilbert spaces and how these spaces generalize the familiar ideas of vector spaces to a more abstract, but incredibly powerful, stage.
Whether you’re a student of linear algebra or just interested in the mathematics behind quantum mechanics, this video connects abstract concepts with practical applications in physics, creating an understanding of both. By the end of this video, you’ll see how these linear algebra concepts come together

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00:00 – Introduction - what are inner product spaces
01:18 – Inner product spaces using usual x-y plane
03:18 – What we mean by "Projections"
05:40 – Extension to higher dimensions - the point of Inner Product Spaces
09:06 – Orthogonality
11:30 – Gram-Schmidt method
24:46 – Hilbert spaces and quantum mechanics