[MUSIC PLAYING] GILBERT STRANG: So,I'm Gilbert Strang.

And this is about my new course18.

065 and the new textbook Linear Algebra andLearning from Data, and what's in those subjects.

So there are reallytwo essential topics and two supplementary, butall very important subjects.

So if I tell you about thosefour parts of mathematics that are in the course,that will give you an idea if you're interestedto follow through.

So the first big subjectis linear algebra.

That subject has just surged,exploded in importance in practice.

What I want to focus on issome of the best matrices, say, symmetric matrices, orthogonalmatrices, and their relation.

Those are the starsof linear algebra.

And the key step isto factor a matrix into maybe symmetrictimes orthogonal matrix, maybe orthogonal times diagonaltimes orthogonal matrix– that's a very importantfactorization called the singular valuedecomposition.

That doesn't get into a lotof linear algebra courses, but it's so critical.

So can I speak now aboutthe second important topic, which is deep learning? So what is deep learning? The job is to create a function.

Your inputs are, likefor driverless cars, the input would be animage that's a telephone pole or a pedestrian.

And the system has to learnto recognize which it is.

Or the inputs fromhandwriting on addresses would be a zip code.

So the system has to learn howto recognize 0, 1, 2, 3 up to 9 from handwriting of all kinds.

Or another one is speech,like what Siri has to do.

So my speech has to getinput and interpreted and output by theprocess of deep learning.

So it involves creatinga learning function.

The function takesthe input, the data, and produces the output,the meaning of that data.

And so what's the function like? That's what mathematicsis about, functions.

So it involves matrixmultiplication.

Part of the functionis multiplying vectors by matrices.

So that's a bunch of steps.

But if there was only that,if it was all linear algebra, the thing wouldfail and has failed.

What makes it work now so muchthat companies are investing enormously in the technologyis that there is now a nonlinear function, a verysimple one in the middle between every pair of matrices.

And that nonlinear function, Ican even tell you what it is.

It's a function f,let's call it f, f of x is equal tox if x is positive.

And f of x is 0if x is negative.

So you can imagine it's graph.

It's a flat linewhere it's negative.

And then it's a 45 degreeslope where it's positive.

So putting thatnonlinear function in between the matrixmultiplications is the way to constructsuccessful learning function.

But you have tofind those matrices.

I mentioned twosupporting subjects.

The first is– optimizationwould be the word.

We have to find the entriesin those matrices that go into the learning function.

That's a crucial step.

So this is a problemof minimizing the error with all those matrixentries as variables.

So this is multivariablecalculus, like 100,000.

500,000 variables, it's justunthinkable in a basic calculus course, but it's happeningin a company that's working with deep learning.

And so that's the giantcalculation of deep learning.

That's what keepsGPU's going for a week.

But it gives amazingresults that could never have been achieved in the past.

So then the other keysubject is statistics.

And the basicideas of statistics play a role here,because when you're multiplying a whole sequenceof matrices in deep learning, it's very possiblefor the numbers to grow out ofsight exponentially or to drop to zero.

And both of those are bad newsfor the learning function.

So you need to keep the meanand variance at the right spot to keep those numbers inthe learning function, those matrices in a good range.

So this course won'tbe a statistics course, but it will use statisticsas deep learning does.

So those are the four subjects.

Linear algebra and deeplearning, two big ones.

Optimization andstatistics, essential also.

So I hope you'll enjoy thevideos and enjoy the textbook.

And go to the OpenCourseWaresite ocw.

Mit.

Edu for the full picture.

Beyond the videos, thereare exercises, problems, discussion, lots more towardmaking a complete presentation, which I hope you like.