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MIT18_06S10_L31

# MIT18_06S10_L31 - 18.06 Linear Algebra Spring 2010...

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18.06 Linear Algebra, Spring 2010 Transcript – Lecture 31 OK. So, coming nearer the end of the course, this lecture will be a mixture of the linear algebra that comes with a change of basis. And a change of basis from one basis to another basis is something you really do in applications. And, I would like to talk about those applications. I got a little bit involved with compression. Compressing a signal, compressing an image. And that's exactly change-of-basis. And then, the main theme in this chapter is th- the connection between a linear transformation, which doesn't have to have coordinates, and the matrix that tells us that transformation with respect to coordinates. So the matrix is the coordinate- based description of the linear transformation. Let me start out with the nice part, which is just to tell you something about image compression. Those of you -- well, everybody's going to meet compression, because you know that the amount of data that we're getting -- well, these lectures are compressed. So that, actually, probably you see my motion as jerky? Shall I use that word? Have you looked on the web? I should like to find a better word. Compressed, let's say. So the complete signal is, of course, in those video cameras, and in the videotape, but that goes to the bottom of building nine, and out of that comes a jumpy motion because it uses a standard system for compressing images. And, you'll notice that the stuff that sits on the board comes very clearly, but it's my motion that needs a whole lot of bits, right? So, and if I were to run up and back up there and back, that would need too many bits, and I'd be compressed even more. So, what does compression mean? Let me just think of a still image. And of course, satellites, and computations of the climate, computations of combustion, the computers and sensors of all kinds are just giving us overwhelming amounts of data. The Web is, too. Now, some compression can be done with no loss. Lossless compression is possible just using, sort of, the fact that there are redundancies. But I'm talking here about lossy compression. So I'm talking about -- here's an image.

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And what does an image consist of? It consists of a lot of little pixels, right? Maybe five hundred and twelve by five hundred and twelve. Two to the ninth by two to the ninth pixels, and so this is pixel number one, one, so that's a pixel. And if we're in black and white, the typical pixel would tell us a gray-scale, from zero to two fifty five. So a pixel is usually a value of one of the xi, so this would be the i-th pixel, is -- it's usually a real number on a scale from zero to two fifty five. In other words, two to the eighth possibilities. So usually, that's the standard, so that's eight -- eight bits. But then we have that for every pixel, so we have five hundred and twelve squared pixels, we're really operating x is a vector in R^n, but what is n? n is five hundred and twelve squared.
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