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Hi. This is the first lecture in MIT's course 18.06, linear algebra, and I'm Gilbert Strang. The text for the course is this book, Introduction to Linear Algebra.

And the course web page, which has got a lot of exercises from the past, MatLab codes, the syllabus for the course, is web.mit.edu/18.06.

And this is the first lecture, lecture one.

So, and later we'll give the web address for viewing these, videotapes. Okay, so what's in the first lecture? This is my plan.

The fundamental problem of linear algebra, which is to solve a system of linear equations.

So let's start with a case when we have some number of equations, say n equations and n unknowns.

So an equal number of equations and unknowns.

That's the normal, nice case.

And what I want to do is — with examples, of course — to describe, first, what I call the Row picture. That's the picture of one equation at a time. It's the picture you've seen before in two by two equations where lines meet.

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

The second picture, I'll put a star beside that, because that's such an important one.

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

And those are the rows and columns of a matrix.

So the third — the algebra way to look at the problem is the matrix form and using a matrix that I'll call A.

Okay, so can I do an example? The whole semester will be examples and then see what's going on with the example.

So, take an example. Two equations, two unknowns. So let me take 2x -y =0, let's say. And -x +2y=3.

Okay. let me — I can even say right away — what's the matrix, that is, what's the coefficient matrix? The matrix that involves these numbers — a matrix is just a rectangular array of numbers. Here it's two rows and two columns, so 2 and — minus 1 in the first row minus 1 and 2 in the second row, that's the matrix.

And the right-hand — the, unknown — well, we've got two unknowns. So we've got a vector, with two components, x and x, and we've got two right-hand sides that go into a vector 0 3.

I couldn't resist writing the matrix form, right — even before the pictures. So I always will think of this as the matrix A, the matrix of coefficients, then there's a vector of unknowns.

Here we've only got two unknowns.

Later we'll have any number of unknowns.

And that vector of unknowns, well I'll often — I'll make that x — extra bold. A and the right-hand side is also a vector that I'll always call b.

So linear equations are A x equal b and the idea now is to solve this particular example and then step back to see the bigger picture. Okay, what's the picture for this example, the Row picture? Okay, so here comes the Row picture.

So that means I take one row at a time and I'm drawing here the xy plane and I'm going to plot all the points that satisfy that first equation. So I'm looking at all the points that satisfy 2x-y =0. It's often good to start with which point on the horizontal line — on this horizontal line, y is zero.

The x axis has y as zero and that — in this case, actually, then x is zero. So the point, the origin — the point with coordinates (0,0) is on the line. It solves that equation.

Okay, tell me in — well, I guess I have to tell you another point that solves this same equation.

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

Then y should be two, right? So there's the point one two that also solves this equation.

And the course web page, which has got a lot of exercises from the past, MatLab codes, the syllabus for the course, is web.mit.edu/18.06.

And this is the first lecture, lecture one.

So, and later we'll give the web address for viewing these, videotapes. Okay, so what's in the first lecture? This is my plan.

The fundamental problem of linear algebra, which is to solve a system of linear equations.

So let's start with a case when we have some number of equations, say n equations and n unknowns.

So an equal number of equations and unknowns.

That's the normal, nice case.

And what I want to do is — with examples, of course — to describe, first, what I call the Row picture. That's the picture of one equation at a time. It's the picture you've seen before in two by two equations where lines meet.

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

The second picture, I'll put a star beside that, because that's such an important one.

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

And those are the rows and columns of a matrix.

So the third — the algebra way to look at the problem is the matrix form and using a matrix that I'll call A.

Okay, so can I do an example? The whole semester will be examples and then see what's going on with the example.

So, take an example. Two equations, two unknowns. So let me take 2x -y =0, let's say. And -x +2y=3.

Okay. let me — I can even say right away — what's the matrix, that is, what's the coefficient matrix? The matrix that involves these numbers — a matrix is just a rectangular array of numbers. Here it's two rows and two columns, so 2 and — minus 1 in the first row minus 1 and 2 in the second row, that's the matrix.

And the right-hand — the, unknown — well, we've got two unknowns. So we've got a vector, with two components, x and x, and we've got two right-hand sides that go into a vector 0 3.

I couldn't resist writing the matrix form, right — even before the pictures. So I always will think of this as the matrix A, the matrix of coefficients, then there's a vector of unknowns.

Here we've only got two unknowns.

Later we'll have any number of unknowns.

And that vector of unknowns, well I'll often — I'll make that x — extra bold. A and the right-hand side is also a vector that I'll always call b.

So linear equations are A x equal b and the idea now is to solve this particular example and then step back to see the bigger picture. Okay, what's the picture for this example, the Row picture? Okay, so here comes the Row picture.

So that means I take one row at a time and I'm drawing here the xy plane and I'm going to plot all the points that satisfy that first equation. So I'm looking at all the points that satisfy 2x-y =0. It's often good to start with which point on the horizontal line — on this horizontal line, y is zero.

The x axis has y as zero and that — in this case, actually, then x is zero. So the point, the origin — the point with coordinates (0,0) is on the line. It solves that equation.

Okay, tell me in — well, I guess I have to tell you another point that solves this same equation.

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

Then y should be two, right? So there's the point one two that also solves this equation.

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