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nHlE7EgJFds | OK, here is lecture
ten in linear algebra. Two important things
to do in this lecture. One is to correct an
error from lecture nine. So the blackboard with that
awful error is still with us. And the second,
the big thing to do is to tell you about
the four subspaces that come with a matrix. We've seen two subspaces,
th... |
bBVDSUZDCWY | last hour we discussed independence and conditional independence and we said that two events that are independent condition dance two events a and B condition than another event C may or may not remain independent when the conditioning is lifted okay this next example shows that conditioning may affect independence oka... |
ZK3O402wf1c | Hi. |
ZK3O402wf1c | This is the first lecture
in MIT's course 18.06, |
ZK3O402wf1c | linear algebra, and
I'm Gilbert Strang. |
ZK3O402wf1c | The text for the
course is this book, |
ZK3O402wf1c | Introduction to Linear Algebra. |
ZK3O402wf1c | And the course web page, which
has got a lot of exercises from |
ZK3O402wf1c | the past, MatLab codes, the
syllabus for the course, |
ZK3O402wf1c | is web.mit.edu/18.06. |
ZK3O402wf1c | And this is the first
lecture, lecture one. |
ZK3O402wf1c | So, and later we'll give the
web address for viewing these, |
ZK3O402wf1c | videotapes. |
ZK3O402wf1c | Okay, so what's in
the first lecture? |
ZK3O402wf1c | This is my plan. |
ZK3O402wf1c | The fundamental problem
of linear algebra, |
ZK3O402wf1c | which is to solve a system
of linear equations. |
ZK3O402wf1c | So let's start
with a case when we |
ZK3O402wf1c | have some number of equations,
say n equations and n unknowns. |
ZK3O402wf1c | So an equal number of
equations and unknowns. |
ZK3O402wf1c | That's the normal, nice case. |
ZK3O402wf1c | And what I want to do is --
with examples, of course -- |
ZK3O402wf1c | to describe, first, what
I call the Row picture. |
ZK3O402wf1c | That's the picture of
one equation at a time. |
ZK3O402wf1c | It's the picture you've
seen before in two |
ZK3O402wf1c | by two equations
where lines meet. |
ZK3O402wf1c | So in a minute, you'll
see lines meeting. |
ZK3O402wf1c | The second picture,
I'll put a star |
ZK3O402wf1c | beside that, because that's
such an important one. |
ZK3O402wf1c | And maybe new to you is the
picture -- a column at a time. |
ZK3O402wf1c | And those are the rows
and columns of a matrix. |
ZK3O402wf1c | So the third -- the algebra
way to look at the problem is |
ZK3O402wf1c | the matrix form and using
a matrix that I'll call A. |
ZK3O402wf1c | Okay, so can I do an example? |
ZK3O402wf1c | The whole semester will be
examples and then see what's |
ZK3O402wf1c | going on with the example. |
ZK3O402wf1c | So, take an example. |
ZK3O402wf1c | Two equations, two unknowns. |
ZK3O402wf1c | So let me take 2x
-y =0, let's say. |
ZK3O402wf1c | And -x 2y=3. |
ZK3O402wf1c | Okay. |
ZK3O402wf1c | let me -- I can even
say right away -- |
ZK3O402wf1c | what's the matrix, that is,
what's the coefficient matrix? |
ZK3O402wf1c | The matrix that involves
these numbers -- |
ZK3O402wf1c | a matrix is just a
rectangular array of numbers. |
ZK3O402wf1c | Here it's two rows and
two columns, so 2 and -- |
ZK3O402wf1c | minus 1 in the first row minus
1 and 2 in the second row, |
ZK3O402wf1c | that's the matrix. |
ZK3O402wf1c | And the right-hand
-- the, unknown -- |
ZK3O402wf1c | well, we've got two unknowns. |
ZK3O402wf1c | So we've got a vector, with
two components, x and y, |
ZK3O402wf1c | and we've got two right-hand
sides that go into a vector |
ZK3O402wf1c | 0 3. |
ZK3O402wf1c | I couldn't resist writing
the matrix form, right -- |
ZK3O402wf1c | even before the pictures. |
ZK3O402wf1c | So I always will think
of this as the matrix A, |
ZK3O402wf1c | the matrix of coefficients,
then there's a vector of |
ZK3O402wf1c | unknowns. |
ZK3O402wf1c | Here we've only
got two unknowns. |
ZK3O402wf1c | Later we'll have any
number of unknowns. |
ZK3O402wf1c | And that vector of
unknowns, well I'll often -- |
ZK3O402wf1c | I'll make that x -- |
ZK3O402wf1c | extra bold. |
ZK3O402wf1c | A and the right-hand
side is also a vector |
ZK3O402wf1c | that I'll always call b. |
ZK3O402wf1c | So linear equations are A
x equal b and the idea now |
ZK3O402wf1c | is to solve this
particular example |
ZK3O402wf1c | and then step back to
see the bigger picture. |
ZK3O402wf1c | Okay, what's the picture for
this example, the Row picture? |
ZK3O402wf1c | Okay, so here comes
the Row picture. |
ZK3O402wf1c | So that means I take
one row at a time |
ZK3O402wf1c | and I'm drawing
here the xy plane |
ZK3O402wf1c | and I'm going to plot
all the points that |
ZK3O402wf1c | satisfy that first equation. |
ZK3O402wf1c | So I'm looking at all the
points that satisfy 2x-y =0. |
ZK3O402wf1c | It's often good to start with
which point on the horizontal |
ZK3O402wf1c | line -- on this horizontal
line, y is zero. |
ZK3O402wf1c | The x axis has y as zero and
that -- in this case, actually, |
ZK3O402wf1c | then x is zero. |
ZK3O402wf1c | So the point, the origin -- |
ZK3O402wf1c | the point with coordinates
(0,0) is on the line. |
ZK3O402wf1c | It solves that equation. |
ZK3O402wf1c | Okay, tell me in -- well,
I guess I have to tell you |
ZK3O402wf1c | another point that solves
this same equation. |
ZK3O402wf1c | Let me suppose x is one,
so I'll take x to be one. |
ZK3O402wf1c | Then y should be two, right? |
ZK3O402wf1c | So there's the point one two
that also solves this equation. |
ZK3O402wf1c | And I could put in more points. |
ZK3O402wf1c | But, but let me put
in all the points |
ZK3O402wf1c | at once, because they all
lie on a straight line. |
ZK3O402wf1c | This is a linear equation
and that word linear |
ZK3O402wf1c | got the letters |
ZK3O402wf1c | Okay, thanks. for line in it. |
ZK3O402wf1c | That's the equation --
this is the line that ... |
ZK3O402wf1c | of solutions to 2x-y=0 my
first row, first equation. |
ZK3O402wf1c | So typically, maybe, x equal
a half, y equal one will work. |
ZK3O402wf1c | And sure enough it does. |
ZK3O402wf1c | Okay, that's the first one. |
ZK3O402wf1c | Now the second one is not
going to go through the origin. |
ZK3O402wf1c | It's always important. |
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