Subspaces are the Natural Subsets of Linear Algebra | Definition + First Examples

A subspace is a subset that respects the two basic operations of linear algebra: vector addition and scalar multiplication. We say they are "closed under vector addition" and "closed under scalar multiplication". On a subspace, you can do linear algebra! Indeed, a subspace is an example of a vector space. We see that all of R^n, {0}, and lines through the origin are all subspaces, but that lines NOT through the origin are not subspaces.

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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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[wei]

This is by far the most elucidated linear algebra course series that I've ever taken.
All my previous teachers were not actually teaching linear algebra, they were just teaching computational rules and never explained the notion or the idea behind it even in a graduate school level, so, it was hard for me to relate linear algebra with other realms of science.

Thank you!

[howtoshotbow]

Thank you! My linear algebra teacher just introduced this topic using theorems and didn’t even explain what it represented. This helped!

[shawcampbell]

Dear education system,

Please make sure your seatbelt is on, the best teachers in the world are now able to teach the entire planet.

It's ridiculous how good Dr Trefor is ; )

[stephen]

This is such a captivating introduction into why we use subspaces (because we now have vectors in our life) and why they must follow the 2 closed rules. Thank you for this great video!

[mustafamarvat]

i have search a lot on youtube and thanks god, finally i got you sir .

[donlansdonlans]

Thanks god I found your channel, you deserve more subs

[sdagupat]

This is absolutely wonderful explanation of vector spaces !!
Man… it clarified so many things for me.

[muyangyan]

This is an awesome explanation! Great visualization.

[nikitathakur]

Thank you sir
It makes the concept more clear and easy

[jingyiwang]

Thank you so much for this amazing explanation about subspaces! It is really helpful!

[rickykim]

such a helpful video that is explained in such an organized, enthusiastic and pedagogical manner! Thanks for the help!!

[bhavneetsingh]

Thank you so much for this wonderful explaination.

[kathirs]

Awesome explanation. Thanks Dr.Trefor !

[rajkumardongre]

Really great explanation, love from India

[umitsedgi]

I love your work! Thanks a lot for all your help here.

[braa]

Thanks your doing the world a service.

[valentinx]

Thank you for this awesome video! <3

[Ravi_Raj_ISM]

thanks a lot sir. also thanks miss ism who shared this video :)

[Realisation]

In the subspace example of a line not passing through the origin, we took the two vectors x and y not belonging to S subspace, but shouldn't we take x and y vectors belonging to S and then check the conditions.

[afiqahjumali]

May i know, what is the application of subspace? 🙏🙏 I hope someone can answer me cause i desperately need the answer