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Hi.

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This is the first lecture in MIT's course 18.06,

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linear algebra, and I'm Gilbert Strang.

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The text for the course is this book,

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Introduction to Linear Algebra.

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And the course web page, which has got a lot of exercises from

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the past, MatLab codes, the syllabus for the course,

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is web.mit.edu/18.06.

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And this is the first lecture, lecture one.

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So, and later we'll give the web address for viewing these,

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videotapes.

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Okay, so what's in the first lecture?

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This is my plan.

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The fundamental problem of linear algebra,

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which is to solve a system of linear equations.

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So let's start with a case when we

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have some number of equations, say n equations and n unknowns.

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So an equal number of equations and unknowns.

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That's the normal, nice case.

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And what I want to do is -- with examples, of course --

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to describe, first, what I call the Row picture.

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That's the picture of one equation at a time.

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It's the picture you've seen before in two

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by two equations where lines meet.

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So in a minute, you'll see lines meeting.

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The second picture, I'll put a star

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beside that, because that's such an important one.

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And maybe new to you is the picture -- a column at a time.

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And those are the rows and columns of a matrix.

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So the third -- the algebra way to look at the problem is

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the matrix form and using a matrix that I'll call A.

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Okay, so can I do an example?

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The whole semester will be examples and then see what's

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going on with the example.

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So, take an example.

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Two equations, two unknowns.

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So let me take 2x -y =0, let's say.

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And -x 2y=3.

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Okay.

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let me -- I can even say right away --

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what's the matrix, that is, what's the coefficient matrix?

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The matrix that involves these numbers --

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a matrix is just a rectangular array of numbers.

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Here it's two rows and two columns, so 2 and --

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minus 1 in the first row minus 1 and 2 in the second row,

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that's the matrix.

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And the right-hand -- the, unknown --

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well, we've got two unknowns.

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So we've got a vector, with two components, x and y,

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and we've got two right-hand sides that go into a vector

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0 3.

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I couldn't resist writing the matrix form, right --

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even before the pictures.

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So I always will think of this as the matrix A,

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the matrix of coefficients, then there's a vector of

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unknowns.

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Here we've only got two unknowns.

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Later we'll have any number of unknowns.

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And that vector of unknowns, well I'll often --

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I'll make that x --

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extra bold.

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A and the right-hand side is also a vector

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that I'll always call b.

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So linear equations are A x equal b and the idea now

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is to solve this particular example

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and then step back to see the bigger picture.

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Okay, what's the picture for this example, the Row picture?

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Okay, so here comes the Row picture.

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So that means I take one row at a time

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and I'm drawing here the xy plane

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and I'm going to plot all the points that

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satisfy that first equation.

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So I'm looking at all the points that satisfy 2x-y =0.

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It's often good to start with which point on the horizontal

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line -- on this horizontal line, y is zero.

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The x axis has y as zero and that -- in this case, actually,

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then x is zero.

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So the point, the origin --

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the point with coordinates (0,0) is on the line.

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It solves that equation.

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Okay, tell me in -- well, I guess I have to tell you

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another point that solves this same equation.

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Let me suppose x is one, so I'll take x to be one.

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Then y should be two, right?

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So there's the point one two that also solves this equation.

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And I could put in more points.

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But, but let me put in all the points

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at once, because they all lie on a straight line.

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This is a linear equation and that word linear

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got the letters

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Okay, thanks. for line in it.

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That's the equation -- this is the line that ...

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of solutions to 2x-y=0 my first row, first equation.

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So typically, maybe, x equal a half, y equal one will work.

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And sure enough it does.

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Okay, that's the first one.

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Now the second one is not going to go through the origin.

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It's always important.

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Do we go through the origin or not?

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In this case, yes, because there's a zero over there.