Change of Basis

We learned about how vectors can form a basis for a vector space, and we can express any vector within a vector space as a linear combination of the basis vectors. But there can be more than one set of basis vectors. What if we want to express a vector using some other basis rather than the standard basis? Well we can implement a change of basis, so let's learn how to do that!
Script by Howard Whittle
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[@jacobharris3002]

This is the simplest explanation of change of basis I have ever seen. I ended up taking two linear algebra courses in college. The first I went through was fine and I was able to apply change of basis easily enough to pass with an A. The second somehow made me more confused than before, probably do to different notation of things and I don't think my professor and textbook were as good. Now I feel like this video has finally cemented the concept and helped me figure out where the equations come from.

[@theaveragepro1749]

this is my current predicament, linear algebra 2 is rough

[@anonymous9217w2]

@@theaveragepro1749 2 is actually way easier than 1, may be u r dumb af

[@rollychairs]

Just in time for my linear algebra exam

[@daylyght3654]

You have no idea how much I do appreciate "this" video! I almost quit linear algebra until I find your video. Thank you so much.

[@madhusudhanannarasimhan9875]

That was one of the simplest videos on this topic that I came across. I am glad I did. You make it easy. Thank you.

[@michaelpisciarino5348]

0:09 Why do we change basis? 0:18 0:25 Recalling previous knowledge of vectors 1:06 Vector Basis Change basis= New Axis, new coordinates 1:52 Writing Vectors with new coordinates in a new basis 2:09 The Same Vector, with different coordinates 3:02 Doing a little Algebra

[@sitproperlywhilewatchingph423]

Thanks man This really helps I wish u a happy life 🙂

[@emilyisreading_]

Thank you so much. I was really confused on change of basis but after watching this I feel way more confident for my linalg final!

[@nirmalkarthik5892]

This video was awesome; it was so clear and thorough!

[@lucynowacki3327]

Nice, but in 6:55 should be plus. Namely, to recover vector in the original basis.

[@kirancp4758]

yes

[@braydenchan138]

I figure out for a while.

[@adityapaul8508]

thanks for clearing that ... i was also a bit confused

[@timothypulliam2177]

I learned more in your video than years of academia. Cheers.

[@ahmedbadday6741]

Thanks professor Dave!

[@bao5860]

6:56 it is clearly U1 + 1/3 U2 , not (-)

[@tauhidanwar3512]

yes it should be U1 + 1/3U2

[@trillonairesclub2329]

no its - remember he got the determinant as - 1/3

[@juergon69]

@@trillonairesclub2329 wHY?

[@anandailyasa2530]

@@trillonairesclub2329 oke if it is -1/3 try subtitute u1 and u2 to the equation and see

[@BoZhaoengineering]

I have to say this is the simplest interpretation of Change of basis. No fancy anime. Just write the simplest math notation and explain it in plain language. Just awesome! The simplest makes the easiest to understand!

[@randymartin5500]

At 6:55 , is the sign correct? Why is there a minus sign when the signs cancelled out during the multiplication of -1/3 . New basis should be V = U1 + 1/3U2

[@rasathashokumar8070]

I think so

[@annamoskalew5465]

There is a mistake for sure. It should be +1/3.

[@sebjornsprauten1406]

yeah i rewatched the video trying to see if i had missed something, but yeah

[@dddhhj8709]

thank you so much .I learned it in such a short time Im suprised

[@dawintch4596]

8:16 Confirmed, Basis changes are all about UWU

[@robin1418]

You deserve at least 10x the number of subscribers.

[@mlworks]

You're awesome Prof.

[@piupolino2618]

7:01, U1 - 1/3 U2 would yield to the vector being 0,1? It would only work if it's + U2

[@user-wr4yl7tx3w]

really well explained

[@davidgruen7423]

7:57 As a mechanic this makes me pumped up!

[@rockyshepheard6054]

Would it be less confusing to write the new u basis eq just like the std ij one? ie. V(arrow) = v sub 1 times m (hat) + v sub 2 times n ) hat)

[@_simply2say]

Cool explanation

[@omerfarooq8223]

Great job man

[@thepleasantcatprincess]

Thank you so much

[@AkimboFennec]

Thanks

[@abdisamad_]

Proff Dave🔥

[@sekharp3305]

Very clear

[@mohdaymanhussainmohdaymanh5574]

Best explanation

[@trillonairesclub2329]

thank you God bless you

[@AriaHarmony]

I feel that doing row reduction using Gaussian elimination is easier, it also works on any matrix including rectangular matrices.

[@dddhhj8709]

🥸 tedious man row reduction

[@vsmn190]

You're incredible.

[@westonley9294]

Very helpful video. Although , I think solving a system of equations is faster than using an inverse matrix for the standard basis.

[@codythurber4060]

You're right... he said it was simpler, but I think he meant to say that it's useful to see that action be made for future topics in LA.

[@just4simplegg428]

Let's change things up ... xD I love this guy!

[@hopewellmuzavazi2365]

Proffesor dave explains for sure

[@aliensoup2420]

I keep getting screwed up - when I'm given a system of vectors, I never know if I'm supposed to use the vectors as rows or columns in the matrix.

[@JimmyCerra]

each vector goes in the column. So the first row are all the x values, second row are the y values, etc... (and so on for higher dimensional vectors).

[@maxpercer7119]

why is it called transition matrix (if we use the inverse transition matrix in practice) ?

[@socialkindle1314]

6:59 shouldn't it be + instead of -1/3u2

[@Imran52Feb]

Here 1, 2 for u1 and vector u2 3, 3 are the I,j coordinates for new bases vectors u1 is a vector (1i 2j )and u2 is a vectorb( 3i 3j) in the i-j plane ? 0ur job here is scaling u1 and u2 so that they form the given vector.

[@sohb143]

I saw that while a change of basis the origin remains the same, what should we do if the origin is also changed, for example, the origin in system 1 is (0,0), but in system 2 it is moved to the coordinates (2,1).

[@luagomesmartins3058]

hi:) you probably dont need this answer anymore haha but im answering for anyone wondering the same. Linear algebra is about a special type of transformations, linear transformations. A transformation is not linear if the origin changes place! (For being linear the origin cant change place and straight lines must stay straight). Hope this helps!

[@adamsonthomas3330]

does this work for R3 and above?

[@mathadventuress]

for the comprehension written in terms of u1= and u2= how did it end up with (16/9)u1+(13/9)u2?

[@zbziyagil]

I think there is a calculation mistake I have found 24/9u1+1/3u2

[@sohamsantra7548]

@@zbziyagil It's correct

[@sasugamegami5113]

4:58 that hit right in the feels

[@pistoletik777]

super! respect

[@OOFtheSecond]

thanks

[@Kevinfreddo]

I am an undergrad chem major who is teaching myself graduate level molecular orbitals theory, and it’s still insane to me how literally all graduate chemistry is linear algebra.

[@asht7788]

U da best

[@shurmurray]

I remember me sitting on my very first lection of linear algebra. 5 minutes later I know how to make a 3D-graphics and was eager to get to my PC =)

[@sriyasomanatham]

can anyone explain the 2nd question in the exercise?

[@yasmeennour47]

+1

[@hazza6265]

isn't it v=u1+1/3*u2 not '-'?

[@ignaciodelavega4631]

4:58 "We have to get rid of you". I knew I was useless, but I think saying that was unecesarry.

[@tempname8263]

Ugh. I had always thought change of basis is a simple operation and it's easy to teach that.

[@BoyaoZhang-me8cg]

I LOVE U!!!!!!!!

[@iboa6938]

what if the basis is a non square matrix

[@maxpercer7119]

the notation is quite convoluted BUT IT MAKES SENSE (sorry caps) - thx professor dave

[@justinsankar1164]

400k!

[@anandailyasa2530]

❤️❤️❤️❤️❤️

[@brd8764]

There are three axis, three dimension and three basis. One is horizontal, other is vertical and third is perpendicular.

[@businesstuber3978]

two years late, but not necessarily; depending on the span, there could potentially be any number of dimensions; this math works for an n-dimensional system.

[@MyHvorfor]

4:55 "In order to isolate this v' term, we have to get rid of YOU" *leaves video in sad desperation*

[@xingzhang4985]

you are my god！

[@kapilprajapat-qb6cz]

6:55 there should be u1 + u2/3

[@aku7598]

Graphically v=u1+(1/3)u2

[@brianbrian4899]

How come no one points out he's wearing exact same T-shirt every single video??

[@ProfessorDaveExplains]

Every series gets a shirt. Check out a different series, see a different shirt.

[@omisingh9677]

HANDS UP IF THE HAIRCUT IS BETTER THEN ORIGINAL

[@jamesa.646]

Thanks for this after watchin' this video I was able to finish my shit Edit: I mean my literal shit, not my homework. I havent started that yet

[@AndrewLyon23]

This just confused me more.

[@plusk343]

bruh dont use v to represent constant because in most books it is used to represent vectors and it can be confusing for beginners to relise

[@irem3450]

i dont understand.bay

[@unrealdevop]

Man, this vector math is insanely hard to grasp. I think it's mostly because all the videos explaining it do so in terms of arbitrary letters rather than real world scenarios....I can't find any decent videos on this in reference to manipulating things inside a 3-dimensional world like a game....for example if I had two actors in a game. One actor is facing another actor and moving towards it (Perhaps at an Angle) and the actor facing that other actors position shifted to the left....how would I determine how far it is now offset from that other actors position if it were still rotated in the same direction....stuff like this seems like it should be elementary stuff and yet I have scoured the internet and not once seen anything remotely close to this in any of the sources I have found.

[@reamabdulsalam524]

Too complicated !!!!