Coordinate Systems From Non-Standard Bases | Definitions + Visualization

We've all used the standard coordinate system where (x,y) means x to the right and y up. However, for any subspace and a basis of that subspace, we can define a coordinate system. The same vector can thus be written in multiple coordinate systems. We describe what exactly we mean by this, introduce some new notation for it (since now we have to keep track of multiple coordinate systems), and visualize it geometrically.

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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.

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This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

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[pencilvulture]

Your videos are an actual godsend with helping me study for tomorrow's linear algebra test. Thank you so much for all the hard work you put into making these top quality videos!!

[radwanalaghawani]

i really appreciate your hard work and it's really nice visualization and you made the idea a piece of cake. your effort deserves more people to see it, but i think if any hade a hard time with the linear aggebra i will immediately give your channel to her/him . thank you again

[tankstocks]

just beautiful! Thank you for making this.

[avinashpandey]

The great lecture i have ever seen on this topic

[jingyiwang]

Thank you so much for making this video! You explain knowledge very clearly.

[priyadarshid]

What kind of app do you use to do the visualization?

[SabrinaXe]

1:40 why is the condition 1) Linear independent has to do with basis vectors not being able to be added to the origin? If basis vectors spans R2 then it should be able to land anywhere on R2 including the origin??

[vijayshankar]

What about the cosine law form, how does it come about ?

[giack]

I've got two questions: when you talk about new basis (b1, b2), the cohordinates of b1 and b2 vectors 
(e.g. b1= (1, 0), b2 = (1, 2)) are still expressed with respect to the standard basis e1 and e2? Or if not so, where these cohordinates come from? Thank you