Change of basis explained simply | Linear algebra makes sense

This video is part of a linear algebra series:

This video is about changing the basis (or coordinate system) of a matrix or a vector. While the change of basis formula is often presented as something to just memorise, we'll see it's actually very very straightforward to understand.

Hints for homework:
1. You can do it! Think about what it does to the basis vectors!

2. Remember, in the original case you first translate the input vector from Bob's basis to Alice's so you can apply M_a, then translate back. A similar idea applies, but you need to translate to something M_a can take as input.

Answers to homework below!!!

Answers
The answer to question one is 2.
The answer to question two is 3.

[scares]

I've seen a few different series on Linear Algebra, and this is definitely the easiest to follow. Thanks so much for making these :)

[InkyAlien]

This series is truly amazing, you're doing an awesome job! Thanks to you after more than a year after I was introduced to linear algebra (by a series of 3blue1brown) I've finally understood the change of basis. You've been explaining everything very clearly and you put emphasis on intuition, which is really important in mathematics in my opinion, and not many people are capable of doing it as well as you do. Thank you for making this series!

[jeongwonkim]

Some professors should get fired and replace with your lecture instead for real. Thank you for these videos !

[vinayakpendse]

I was doing linear algebra till now mechanically just applying formulae everywhere, but your videos really helped me.

[Doggyshakespeare]

I think having the multiple choice questions built in the video is an awesome idea, because it gets us to actively think about and apply what is being taught and to gauge how much we really understand. Instead of passively watching and possibly forgetting the content soon afterwards, it helps us retain the information. Thanks for the videos!

[andrewniven]

Glad to see there are no dislikes for this video - anyone who tries to explain anything to do with vectors is a hero of mine :-)

[Ennar]

As someone who teaches mathematics at university level (to math majors), I think you are doing quite fine. I (like to) think that most of my colleagues who teach math majors do actually explain this stuff, but probably in other STEM fields, due to the lack of time to cover it, lots of these intuitive explanations get skipped over. So, please keep making these videos, they are becoming a good, casual, reference to the subject. You are a huge help!

[AndrewTaylorPhD]

See, this is what I love about this channel — it's just the same attitude I've always had. I can understand something and work out the formula when I need it much more easily than I can memorise a formula, and about half of my professors at uni just didn't get that. I flunked a whole class on calculus because I couldn't internalise the maths until a friend described it in terms I understood and it clicked.

[nataliarodriguez]

This series made me literally understand all the unconnected facts i had from my uni course and it finally made SO MUCH SENSE. It makes so much sense it made me have my own amazing insights over lots of things you haven't covered!!

[russsulit]

From the time I'm writing this comment I have yet to do an assignment due in 3 days. I haven't watched lectures (hard to follow), I'm out of hope and keep questioning my existence but this video has given me a slight glimmer, a slight chance. Thank you

[dlseller]

This is the best explanation of change of basis I've seen. Ever text book I've read talks about changing one vector into another. This is the first time I've seen someone say that it's the same vector described a different way. ( which is what change of basis is really about).
Thank you!

[physicsguy]

This is how I was taught to think about linear algebra by my great prof in undergrad, and ever since, I've felt like I've had a leg up on everyone who didn't take the time to think about the subject as much or didn't have him as a prof. Seriously, I still run into people in grad school (currently a physics grad student) who struggle on certain problem sets because they never were taught to think about linear algebra the "right" way years ago. I just wanted to share this so that other viewers know how lucky they are to have your videos. Really great stuff!

[tensorbundle]

Woman in mathematics.! Really appreciate. Keep up the good work

[paullivi]

Keep uploading your extremely interesting videos!!👏🏻

[danielkunigan]

I absolutely hated matrices and vectors before your series, now I've found myself playing around with them voluntarily. These are great videos, thanks for putting in the effort to make them.

[Theox]

5:49 _"usually a matrix takes a vector and transforms it to some other vector"_
The matrix takes actually a coordinate vector of some vector v and transforms it to the coordinate vector of f(v) for some linear map f (represented by the matrix). Note that the coordinate vectors of v and f(v) can be with respect to different bases, even if f maps to the same vector space. This becomes important when we want to understand that the matrix Q is a "real" matrix after all, it is the matrix representing the identity map with respect to different bases for domain and target (which are both the same space).

[TheViolaBuddy]

I really love this series; I'm not sure if there's too much more to say. I'm sure you're getting lots of comparison to 3B1B's linear algebra series, but it's always good to see things explained multiple times in multiple ways, especially since this series has an underlying quantum mechanical slant to it that makes it different from that series (and I can't place my finger on exactly why I feel that way - maybe it's just a psychological bias that I _know_ you're going to be linking back to quantum mechanics so I'm more alert for references, or maybe there's something really there, like the focus on vectors "really" being just a linear combination of basis vectors, or something). Either way, always looking forward to future videos!

[vaibhavjindal]

It's surprising how concepts like this taught stupidly in class room can be taught in such a lucid fashion. Good Job.

[Packbat]

Belated reply to the third homework question: I knew some linear algebra before, but this was still informative and also really inspiring. The problems were all straightforward in a good way, and I was utterly delighted when I realized you stuck an information-destroying transform into the initial video on transforms to prepare everyone for when matrices with no left-inverse turned up later on.

Speaking of which: the concept of a left-inverse was incredibly clarifying. Thank you for including that.

[shethnisarg]

This video literally makes the concept crystal clear! Thank you very much!!