Lengths, angles, projection, correlation | Linear algebra episode 2

#vectors #linearalgebra #dotproduct #innerproductspace #orthogonality #correlation #fourier_series

The dot product has many cool applications in artificial intelligence, audio engineering, and classical mechanics. It measures the correlation between different pieces of data, which allows AI algorithms to discover new concepts. It also measures lengths and distances so that we can find out how closely one sound wave approximates another. We also look at a few weird and exotic inner products such as the taxicab distance.

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Here are many extra links you can follow if you want to dig deeper:

[WIKI 1] en.wikipedia.org/wiki/Dot_product
Here you can find the higher-dimensional proof for the equivalence between the two formulas for the dot product.

[3B1B 1]    • Dot products and duality | Chapter 9, Esse...  
This video explains the dot product using the concept of dual vectors. We will talk about those later when we cover tensor algebra.

[ZS 1]    • The real world applications of the dot pro...  
Some cool real-world applications of inner products. One of them tries to find similarities between documents, by encoding all words as vectors first.

[TB 1]    • The geometric view on orthogonal projections  
Derives the orthogonal projection formula. Then applies it to the standard basis vectors, to show that it retrieves the coordinates of a vector.

[TB 2]    • Least Squares Approximations  
   • Reducing the Least Squares Approximation t...  
Uses dot products to find the closest solution to a linear equation.

[MTB 1]    • Linear Algebra 20L: The Dot Product in Non...  
Expressing the dot product in a non-orthogonal basis. This leads to the very important concept of a metric, which we will encounter again in future videos on All Angles.

[MTB 2]    • Introducing the Inner Product - A Fundamen...  
This is a very good in-depth series dedicated to inner products. It tackles many related topics such as symmetric matrices, metrics, Fourier analysis, Legendre polynomials, Gaussian integration techniques, and much more. Pavel Grinfeld is one of the best teachers on the web.

0:00 Why do we need the dot product?
2:15 Some exotic examples of inner products
3:49 Definition of the dot product
5:12 Length/norm/modulus
8:42 Measuring angles
9:48 Dot product not always possible
14:04 Orthogonality & correlation
20:40 You cannot divide vectors (yet)
21:53 Projections
23:48 The projection formula
25:37 Using projections for approximations
26:34 Applications of projections in music & physics
28:17 Proof that the two formulas are equal
29:48 Rules for inner products

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