Vector Spaces | Definition & Examples

Vectors in R^n obey a list of rules, things like commutivity of vector addition that a+b = b+a as vectors. However, we can abstract this list of rules and introduce the general concept of a vector space. A vector space has elements that might be all kinds of weird things, but as long as they obey the same list of rules, they are a vector space. And the key here is that all the things we have done in linear algebra extend to this new situation too! Our first new example is polynomial of degree less than or equal 2, which have a natural "vector" addition and scalar multiplication. ************************************************** 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:    • How to watch math videos   ►5 Tips to Make Math Practice Problems Effective:    • 5 Tips To Make Math Practice Problems Actu...   **************************************************** ►Want some cool math? Check out my "Cool Math" Series:    • Cool Math Series   **************************************************** Course Playlists: ►Calculus I:    • Calculus I (Limits, Derivative, Integrals)...   ►Calculus II:    • Older Calculus II (New Playlist For Spring...   ►Discrete Math:    • Discrete Math (Full Course: Sets, Logic, P...   ►Linear Algebra:    • Linear Algebra (Full Course)   ***************************************************** ►Follow me on Twitter:   / treforbazett   ***************************************************** This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati. BECOME A MEMBER: ►Join:    / @drtrefor   MATH BOOKS & MERCH I LOVE: ► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett