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
Angles and Orthogonality: • Angles and Orthogonali...
Linear Algebra course: • Linear Algebra
Linear Algebra exercises: • Linear Algebra Exercises
Join Wrath of Math to get exclusive videos, music, and more:
/ @wrathofmath
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◉
Graph Theory: amzn.to/3JHQtZj
Real Analysis: amzn.to/3CMdgjI
Abstract Algebra: amzn.to/3IjoZaO
Linear Algebra: amzn.to/43xAWEz
Calculus: amzn.to/3PieD1M
Proofs and Set Theory: amzn.to/367VBXP (available for free online)
Statistics: amzn.to/3tsaEER
Discrete Math: amzn.to/3qfhoUn
Number Theory: amzn.to/3JqpOQd
★DONATE★
◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: / wrathofmathlessons
◆ Donate on PayPal: www.paypal.me/wrathofmath
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...
● Instagram: / wrathofmathedu
● Facebook: / wrathofmath
● Twitter: / wrathofmathedu
My Math Rap channel: / @mathbars2020