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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[jacobharris]

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.

[daylyght]

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.

[michaelpisciarino]

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

[rollychairs]

Just in time for my linear algebra exam

[lucynowacki]

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

[madhusudhanannarasimhan]

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.

[dawintch]

8:16 Confirmed, Basis changes are all about UWU

[bao]

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

[randymartin]

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

[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!

[timothypulliam]

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

[robin]

You deserve at least 10x the number of subscribers.

[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!

[davidgruen]

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

[yogitajindal]

Great explanation Thank you so much 🙏🏻

[piupolino]

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

[nirmalkarthik]

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

[socialkindle]

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

[AriaHarmony]

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

[aliensoup]

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.