Visualizing Diagonalization & Eigenbases

Diagonal transformations are really nice to visualize geometrically. In 2D they are just a combination of horizontal and vertical stretching. While a generic matrix isn't quite this nice, if you can find a basis of eigenvectors, then the transformation "looks" like stretching and compressing along those eigenvectors by the values of the eigenvectors. This makes it pretty nice, but we can do better. When we diagonalize a matrix, this is a composition of transformations. You first apply a change-of-basis to convert from the standard basis to the eigenbasis. Then you apply the nice diagonal transformation. Finally, you convert back. ************************************************** Now it's your turn: 1) Summarize the big idea of this video in your own words 2) Write down anything you are unsure about to think about later 3) What questions for the future do you have? Where are we going with this content? 4) Can you come up with your own sample test problem on this material? Solve it! Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK. *************************************************** Want more ideas for learning math effectively? ►How to Watch Math Videos: https://www.youtube.com/watch?v=Ljq5y3ksk8o ►5 Tips to Make Math Practice Problems Effective: https://www.youtube.com/watch?v=LPH2lqis3D0 **************************************************** ►Want some cool math? Check out my "Cool Math" Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho **************************************************** Course Playlists: ►Calculus I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ►Calculus II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdQCBjYswqbn7LxL1pW4cW4 ►Discrete Math: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►Linear Algebra: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 ***************************************************** ►Follow me on Twitter: http://twitter.com/treforbazett ***************************************************** This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati. BECOME A MEMBER: ►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join MATH BOOKS & MERCH I LOVE: ► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett