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, ) 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 I could put in more points. But, but let me put in all the

points at once, because they all lie on a straight

line. This is a linear equation and that word linear got the letters for

line in it. That's the equation -- this is the line that ... of

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

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

work. And sure enough it does. Okay, that's the first one. Now

the second one is not going to go through the origin. It's

always important. Do we go through the origin or not?

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

In this case we don't go through the origin, because if x and y are

zero, we don't get three. So, let me again say suppose y is

zero, what x do we actually get? If y is zero, then I get x is minus

three. So if y is zero, I go along minus three. So there's

one point on this second line. Now let me say, well, suppose x is

minus one -- just to take another x. If x is minus one, then this is a

one and I think y should be a one, because if x is minus one,

then I think y should be a one and we'll get that point.

Is that right? If x is minus one, that's a one. If y is a one, that's

a two and the one and the two make three and that point's

on the equation. Okay. Now, I should just draw the

line, right, connecting those two points at -- that will give me the

whole line. And if I've done this reasonably well,

I think it's going to happen to go through -- well,

not happen -- it was arranged to go through that point.

So I think that the second line is this one, and this is the

all-important point that lies on both lines. Shall we just check

that that point which is the point x equal one and y was two,

right? That's the point there and that, I believe, solves

both equations. Let's just check this.

If x is one, I have a minus one plus four equals three,

okay. Apologies for drawing this picture that you've seen before.

But this -- seeing the row picture -- first of all, for n equal 2,

two equations and two unknowns, it's the right place to start. Okay.

So we've got the solution. The point that lies on both lines.

Now can I come to the column picture? Pay attention,

this is the key point. So the column picture.

I'm now going to look at the columns of the matrix.

I'm going to look at this part and this part. I'm going to say that

the x part is really x times -- you see, I'm putting the two -- I'm

kind of getting the two equations at once --

that part and then I have a y and in the first equation it's multiplying

a minus one and in the second equation a two,

and on the right-hand side, zero and three. You see, the

columns of the matrix, the columns of A are here and the

right-hand side b is there. And now what is the equation asking

for? It's asking us to find -- somehow to combine that vector and

this one in the right amounts to get that one. It's asking us to find

the right linear combination -- this is called a linear combination.

And it's the most fundamental operation in the whole course.

It's a linear combination of the columns. That's what we're seeing

on the left side. Again, I don't want to write down a

big definition. You can see what it is.

There's column one, there's column two.

I multiply by some numbers and I add. That's a combination --

a linear combination and I want to make those numbers the right numbers

to produce zero three. Okay. Now I want to draw a picture

that, represents what this -- this is algebra.

What's the geometry, what's the picture that goes with it?

Okay. So again, these vectors have two components,

so I better draw a picture like that. So can I put down these columns?

I'll draw these columns as they are, and then I'll do a combination of

them. So the first column is over two and down one, right?

So there's the first column. The first column. Column one.

It's the vector two minus one. The second column is --

minus one is the first component and up two. It's here.

There's column two. So this, again, you see what its

components are. Its components are minus one,

two. Good. That's this guy. Now I have to take a combination.

What combination shall I take? Why not the right combination, what

the hell? Okay. So the combination I'm going

to take is the right one to produce zero three and then we'll see it

happen in the picture. So the right combination is to take

x as one of those and two of these. It's because we already know that

that's the right x and y, so why not take the correct

combination here and see it happen? Okay, so how do I picture this

linear combination? So I start with this vector that's

already here -- so that's one of column one,

that's one times column one, right there. And now I want to add on --

so I'm going to hook the next vector onto the front of the arrow will

start the next vector and it will go this way. So let's see,

can I do it right? If I added on one of these vectors,

it would go left one and up two, so we'd go left one and up two, so

it would probably get us to there. Maybe I'll do dotted line for that.

Okay? That's one of column two tucked onto the end,

but I wanted to tuck on two of column two. So that --

the second one -- we'll go up left one and up two also.

It'll probably end there. And there's another one. So what

I've put in here is two of column two.

Added on. And where did I end up? What are the coordinates of this

result? What do I get when I take one of this plus two of that?

I do get that, of course. There it is, x is zero, y is three, that's b.

That's the answer we wanted. And how do I do it? You see I do

it just like the first component. I have a two and a minus two that

produces a zero, and in the second component I have a

minus one and a four, they combine to give the three.

But look at this picture. So here's our key picture.

I combine this column and this column to get this guy.

That was the b. That's the zero three. Okay. So that idea of

linear combination is crucial, and also -- do we want to think

about this question? Sure, why not. What are all the combinations?

If I took -- can I go back to xs and ys? This is a question for

really -- it's going to come up over and over, but why don't we see it

once now? If I took all the xs and all the ys, all the combinations,

what would be all the results? And, actually,

the result would be that I could get any right-hand side at all.

The combinations of this and this would fill the whole plane.

You can tuck that away. We'll, explore it further. But this idea

of what linear combination gives b and what do all the linear

combinations give, what are all the possible,

achievable right-hand sides be -- that's going to be basic. Okay.

Can I move to three equations and three unknowns?

Because it's easy to picture the two by two case.

Let me do a three by three example. Okay, I'll sort of start it the

same way, say maybe 2x-y and maybe I'll take

no zs as a zero and maybe a -x+2y and maybe a -z as a --

oh, let me make that a minus one and, just for variety let me take,

-3z, -3ys, I should keep the ys in that line, and 4zs is, say, 4.

Okay. That's three equations. I'm in three dimensions, x, y,

z. And, I don't have a solution yet. So I want to understand the

equations and then solve them. Okay. So how do I you understand

them? The row picture one way. The column picture is another very

important way. Just let's remember the matrix form,

here, because that's easy. The matrix form -- what's our matrix A?

Our matrix A is this right-hand side, the two and the minus one and

the zero from the first row, the minus one and the two and the

minus one from the second row, the zero, the minus three and the

four from the third row. So it's a three by three matrix.

Three equations, three unknowns. And what's our right-hand side?

Of course, it's the vector, zero minus one, four.

Okay. So that's the way, well, that's the short-hand to write

out the three equations. But it's the picture that I'm

looking for today. Okay, so the row picture.

All right, so I'm in three dimensions, x,

y and z. And I want to take those equations one at a time and ask --

and make a picture of all the points that satisfy --

let's take equation number two. If I make a picture of all the

points that satisfy -- all the x, y, z points that solve

this equation -- well, first of all,

the origin is not one of them. x, y, z -- it being 0, 0, 0 would

not solve that equation. So what are some points that do

solve the equation? Let's see, maybe if x is one,

y and z could be zero. That would work, right? So there's one point.

I'm looking at this second equation, here, just, to start

with. Let's see. Also, I guess,

if z could be one, x and y could be zero,

so that would just go straight up that axis. And,