Inner Products and Inner Product Spaces | Linear Algebra

We introduce inner products and inner product spaces. We'll see how the inner product axioms come directly from the basic properties of the dot product, we'll briefly discuss complex inner product spaces and the symmetry axiom for such spaces, we'll see how norms and distances are defined and calculated in inner product spaces, see examples and non-examples of inner product spaces including an inner product space of matrices, and will see basic algebraic properties of inner products. #linearalgebra

Join Wrath of Math to get exclusive videos, music, and more:

0:00 Intro
1:29 Definition of Inner Product and Inner Product Space
4:26 Euclidean n-Space (with Dot Product)
4:58 Norm and Distance in Inner Product Space
5:45 Properties of Norm and Distance
6:31 Inner Product Non-Example
8:23 Weighted Euclidean Inner Product
11:59 Norm and Distance in Weighted Euclidean Inner Product Space
14:24 Algebraic Properties of Inner Products
17:42 Standard Inner Product on Square Matrices
20:08 Conclusion

◉Textbooks I Like◉

★DONATE★

Thanks to Loke Tan, Raül Beienheimer, Matt Venia, Micheline, Doug Walker, Odd Hultberg, Marc, Shlome Ashkenazi, Barbora Sharrock, Mohamad Nossier, Rolf Waefler, Shadow Master, and James Mead for their generous support on Patreon!

Outro music is mine. You cannot find it anywhere, for now.

Follow Wrath of Math on...

[imora]

I'm not one to comment on videos, but this was amazing, thank you! You should have way more likes!!

[MatthewBricker]

As a math major, your videos are an invaluable resource. Thank you!

[PunmasterSTP]

A bit off-topic, but has anyone else ever been to Inner Space Cavern? 😃