Change of Basis

MIT 18.06SC Linear Algebra, Fall 2011
Instructor: Martina Balagovic

A teaching assistant works through a problem on change of basis.

License: Creative Commons BY-NC-SA

[weizong]

This recitation seems to be a glimpse of a more advanced linear algebra course. It contains a lot of knowledge that we do not know yet.

[yangpiao]

I think the key to understand the whole thing is that w1 w2 w3 form a basis of polynomial, so wi must be a combination of 1 x and x squared.

[nerophon]

I found the phrasing of the problem confusing. To me, the table implies that, for any polynomial in the space, alpha = 1 when x = -1. But this is impossible; hence the confusion. I was also surprised by the choice of the differentiation matrix D later in the explanation; the natural choice for me would be the transpose of what was written, so that Dx = d/dx where x and d/dx are column vectors.

[krischalkhanal]

I think understanding this is the real deal. I tried by my own brute force way, and it did work, but she is using shortcuts here, which I am having hard time wrapping my head around. Much struggle needed.

[BluesbreakerWyatt]

If you find this difficult, it helps a lot to read related chapters in the book "Introduction to Linear algebra", it cover much more details.

[tetsuroda]

7:33 I do not understand why the matrix changes bases as required. The reason why I get confused is probably because w_{i} is not an element but a vector. Can somebody make it clearer?

[MikhailSamin]

This is absolutely cursed, I struggled for like half an hour to understand the problem and why is the polynomial at some point is some specific other polynomial, like, ???

The most cursed problem in all the recitations

[NisargJain]

6:32 how did she conclude that?
can somebody elaborate that?


Thanks.

[quirkyquester]

what's happening at 6:36 my lorrrdd, please save meeee oops

[Longpan]

Nice exercise and good job, though maybe a bit fast in the explanation. But eventually, using the various comments, I fall back on my 2 legs. Thanks 🙏🏻

[dalisabe]

Not too clear and I am sure the concept should be a fairly easy one explained in more organized steps.

[iamjojo]

Good polynomial examples, but poor explanations.
She just go through it, not explain a bit.
I can't understand at all until I see comments.
She should have written down w1, w2, w3 in polynomial form first.
Namely, assume w1=a1×(1)+b1×(x)+ c1× (x^2)
And w2=a2×(1)+b2×(x)+ c2× (x^2)
And w3=a3×(1)+b2×(x)+ c3× (x^2)
So, written above 3 equations into matrix form
We have [w1(next row) w2 (next row) w3]=
[a1, b1, c1 (next row) a2, b2, c2 (next row) a3, b3, c3 ] × [ 1 (next row) x (next row) x^2]
By writting in matrix form, we can easily observe the 3 by 3 matrix of a1 b1 c1 a2 b2 c2 a3 b3 c3 is the matrix that change basis of right hand side (1, x, x^2) to basis of left hand side (w1, w2, w3)
So all we need to do is inserting known values of x=-1, 0, 1 and corresponding w1, w2, w3 at these points into above matrix to get
all the coeeficients, which is what she does in part b.

[pydro]

entry 2, 1 of A inverse is 1/2 (not -1/2)
thanks

[thedailyepochs]

i think khan academy does a better job at explaining basis than this recitation. Good lecture though