The Span is a Subspace | Proof + Visualization

The Span of two vectors - ie all linear combinations of them - turns out to be a subspace. That is, it is closed under scalar multiplication, vector addition, and contains the zero vector. Geometrically, it sweeps out a plane (except if the vectors are multiplies of one another or the zero vector).
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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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[@alexstec6826]

Your work is unbelievable. Funny to think professors who have been teaching this stuff for years can't explain as well as you do.

[@shiledouglas556]

Truth

[@ryanbayer2443]

thank god tref is my prof

[@andyhype2546]

What a legend. May your children be many and plentiful.

[@notquite6foot2]

Excellent job combining the visuals with the content -- makes it easier to understand what's going on! Great production value. Thank you for making these courses! It's what's getting me through linear algebra this summer. Also, the order you present the information in is really helpful. "Why would we want to do that? I'm going to argue ..." and "Let's look at a few pathological/extreme examples" -- it's clear you've answered lots of questions from students, and thought through how to get from concept A to concept B.

[@codemathlab]

Such an amazing explanation. I haven't seen an explanation like this before on this topic.

[@warisulimam3440]

Just wanted to express my appreciation for the tremendously helpful content you create, professor. Super grateful to you!

[@jameskennedy674]

This is really good stuff. Explanations are very easy to follow and the video puts a lot of things into perspective.

[@dallasdominguez9501]

Great explanation man. Just what I needed

[@technicaljethya993]

I am just looking for a channel that has Short videos with visualize then i found you🤩🤩

[@no-qm1kn]

Incredible job! Thanks so much

[@angelaagullana1751]

Awesome videos! Made it easier to understood the lectures I've been spending hours trying to understand because I couldn't visualize it. Thanks!

[@DrTrefor]

Thank you! Good visualization is so important imo

[@rawalrohit7929]

great explanation. great visual. loved it

[@njoy2075]

You put a lot of effort in your videos.. clearly explains the mysteries of the maths.. you do deserve way more credit than this

[@njoy2075]

@@DrTrefor Can you please teach us about Tridiagonal Toeplitz Matrices

[@flameprincess7313]

This is great, I just wish you had used actual constants and vectors as well for an example :)

[@mortezanaghavimoghadam8258]

really good explanation thanks

[@shrikantprabhu799]

You are a very good teacher ! Great effort to make Linear algebra simple :) ... God bless !

[@DrTrefor]

Glad you think so!

[@kamogelomorole256]

Thanks you 🤘🏾

[@VolumetricTerrain-hz7ci]

There are unknown way to visualize subspace, or vector spaces. You can stretching the width of the x axis, for example, in the right line of a 3d stereo image, and also get depth, as shown below. L R |____| |______| This because the z axis uses x to get depth. Which means that you can get double depth to the image.... 4d depth??? :O p.s You're good teacher!

[@Ravi_Raj_ISM]

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

[@Adam-gw8jt]

Can somebody explain why proving the 3 cases prove that the span must be a subspace? Like I don't understand the relationship between it.

[@cal8391]

vivid

[@maraj4563]

name of the software you use in digram?

[@DrTrefor]

Matlab!

[@CanDoSo_org]

span{a,b} is a 2-d plane, how about span{a, b, c}? The whole 3-d space?

[@DrTrefor]

Yes, IF they are linearly independent.

[@CanDoSo_org]

@@DrTrefor Thanks, Dr.

[@andyhype2546]

Are you going to watch the Shakira Bowl?

[@maxpercer7119]

yes