Lec 16 | MIT 18.06 Linear Algebra, Spring 2005

[Ondelendo]

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[RickieDeanzc92]

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[akhil089]

thanks professor gilbert

[dylanhoggyt]

@pelemanov from what I understand we are projecting b in to the column space so that we can actually solve the system with a best estimate. The closest (least squares) projection of b is p which will only be orthogonal if b happens to be in the null space of A transpose, but there is no requirement that this is the case. In fact if b happens to be in the column space, then the projection doesn't change b at all (i.e. P = I.)

[pelemanov]

@dylanhoggyt I get that, but that doesn't answer my question. What I mean, is that around minute 14:00 he says that the error is vertical instead of orthogonal to the line. I thought we were trying to minimize the error by orthogonal projection. I'm probably mixing things up, but I don't see it.

[j4ckjs]

@pelemanov by the way, you can check out Lec 24b, which is the review lecture. And he mentioned this "two pictures" of view.

[pelemanov]

Great lecture! But aren't we supposed to make an orthogonal projection? Instead he did a projection parallel to the Y-axis because he calculates p1, p2 and p3 by taking t-values 1, 2 and 3. You can also see it on his drawing. Why does he take this projection instead of the orthogonal one? And how can e turn out to be orthogonal to p anyway?

[j4ckjs]

@pelemanov Not sure if I understand your question. You mean why he solve A^tAx = A^t b instead of Ax = Pb? The reason is that they are the same! Expand P and manipulate the terms then you'll see.

[pelemanov]

@j4ckjs What I mean, is that around minute 14:00 he says that the error is vertical instead of orthogonal to the line. I thought we were trying to minimize the error by orthogonal projection. I'm probably mixing things up, but I don't see it.

[j4ckjs]

@pelemanov Because b, p and e are 3dimensional vectors. Actually things are happening in the 3d space.

[asdfghjklouise]

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[buzzingbear]

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[gladragsakshay]

Prof Strang spoke so much about errors and he did make one ! :P

[hoofhearted1212]

" so what's up?" lol

[tekerirem]

I am tired of seeing stupid featured videos at the top.

[jaryH3]

What was he doing since 22:45? I don't see why was he augmenting.

[666modac1]

@jaryH3 He just combined [A]^T[A] on the left side with [A]^T[b] on the right side to make [A]^T[A|b].

[heropadaimazero]

The mistake starts at 11:26. The right hand side is (1 2 2), not (1 2 3). But he later uses the (1 2 2) for all other calculations, so not a big deal.

[Impresioniste]

I mean would be if the b column where 123. Sorry about my english.

[Impresioniste]

An error on the b column: is 1,2,2 not 1,2,3 because in this case the solution will be in the column space because there is a column 1,2,3.