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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.
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