Abstract Linear Algebra 4 | Basis, Linear Independence, Generating Sets

📝 Access all videos and PDFs: https://tbsom.de/s/ala 👍 Become a member on Steady: https://steadyhq.com/en/brightsideofm... 👍 Or become a member on Patreon:   / bsom   Other possibilities here: https://tbsom.de/sp You can also support me via PayPal: https://paypal.me/brightmaths Or via Ko-fi: https://ko-fi.com/thebrightsideofmath... Or via Patreon:   / bsom   Or via other methods: https://thebrightsideofmathematics.co... Join this channel on YouTube:    / @brightsideofmaths   💬 Access to the community forum: https://thebrightsideofmathematics.co... 🕚 Early access for videos: https://thebrightsideofmathematics.co... ❓ FAQ: https://thebrightsideofmathematics.co... 🛠️ What tools do you use: https://thebrightsideofmathematics.co... 📚 Download my books: https://thebrightsideofmathematics.co... 🆓 Ad-free access to all videos: https://thebrightsideofmathematics.co... ▶️ Exclusive supporter videos: https://tbsom.de/s/ala 👏 Your name at the top in the credits of the upcoming videos! (opt-out possible) 📝 PDF versions, quizzes, and Python scripts: https://tbsom.de/s/ala Please consider to support me if this video was helpful such that I can continue to produce them :) As a supporter you can access the videos directly from my website. If you need more information, just send me an email: https://tbsom.de/s/mail Watch the whole video series about Abstract Linear Algebra and download PDF versions and quizzes: https://tbsom.de/s/ala Supporting me via Steady is the best option for me and you. Please consider choosing a supporter package here: https://tbsom.de/s/subscribe 🌙 There is also a dark mode version of this video:    • Abstract Linear Algebra 4 | Basis, Linear ...   🔆 There is also a bright mode version of this video:    • Abstract Linear Algebra 4 | Basis, Linear ...   🔆 To find the YouTube-Playlist, click here for the bright version:    • Abstract Linear Algebra   🌙 And click here for the dark version of the playlist:    • Abstract Linear Algebra [dark]   🙏 Thanks to all supporters! They are mentioned in the credits of the video :) This is my video series about Abstract Linear Algebra. It extends the Linear Algebra series by considering more general vector spaces. We will discuss abstract vector spaces, abstract inner products, the change-of-basis matrix,different bases, polynomial spaces, and so on. I hope that it will help everyone who wants to learn about it. For any questions, please leave a comment or come to the community forum of the Bright Side of Mathematics: https://tbsom.de/s/community 00:00 Introduction 00:36 Definition of polynomial spaces 02:27 Definition: linear combination 03:38 Definition: span of a subspace 04:44 Definition: generating set for a subspace 05:29 Definition: linear independent sets 06:36 Definition: basis of a subspace 07:57 Definition: dimension of a subspace 09:18 Examples for polynomial spaces 11:25 Example for infinite-dimensional vector space 11:44 Example for matrices 12:42 Credits #LinearAlgebra #FunctionalAnalysis #Mathematics #LearnMath #calculus #vectors I hope that this helps students, pupils and others. Have fun! (This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on) The Bright Side of Mathematics has whole video courses about different topics and you can find them here https://tbsom.de/s/start For questions, you can contact me: https://steadyhq.com/en/backend/messa...