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ZK3O402wf1c | How could it go wrong
that out of these -- |
ZK3O402wf1c | out of three columns and
all their combinations -- |
ZK3O402wf1c | when would I not be able
to produce some b off here? |
ZK3O402wf1c | When could it go wrong? |
ZK3O402wf1c | Do you see that
the combinations -- |
ZK3O402wf1c | let me say when it goes wrong. |
ZK3O402wf1c | If these three columns
all lie in the same plane, |
ZK3O402wf1c | then their combinations
will lie in that same plane. |
ZK3O402wf1c | So then we're in trouble. |
ZK3O402wf1c | If the three columns
of my matrix -- |
ZK3O402wf1c | if those three vectors happen
to lie in the same plane -- |
ZK3O402wf1c | for example, if
column three is just |
ZK3O402wf1c | the sum of column one and column
two, I would be in trouble. |
ZK3O402wf1c | That would be a matrix A
where the answer would be no, |
ZK3O402wf1c | because the combinations -- |
ZK3O402wf1c | if column three is in the same
plane as column one and two, |
ZK3O402wf1c | I don't get anything
new from that. |
ZK3O402wf1c | All the combinations are in the
plane and only right-hand sides |
ZK3O402wf1c | b that I could get would
be the ones in that plane. |
ZK3O402wf1c | So I could solve it for
some right-hand sides, when |
ZK3O402wf1c | b is in the plane, but
most right-hand sides |
ZK3O402wf1c | would be out of the
plane and unreachable. |
ZK3O402wf1c | So that would be
a singular case. |
ZK3O402wf1c | The matrix would
be not invertible. |
ZK3O402wf1c | There would not be a
solution for every b. |
ZK3O402wf1c | The answer would
become no for that. |
ZK3O402wf1c | Okay. |
ZK3O402wf1c | I don't know -- |
ZK3O402wf1c | shall we take just a
little shot at thinking |
ZK3O402wf1c | about nine dimensions? |
ZK3O402wf1c | Imagine that we have vectors
with nine components. |
ZK3O402wf1c | Well, it's going to be
hard to visualize those. |
ZK3O402wf1c | I don't pretend to do it. |
ZK3O402wf1c | But somehow, pretend you do. |
ZK3O402wf1c | Pretend we have -- if this
was nine equations and nine |
ZK3O402wf1c | unknowns, then we would
have nine columns, |
ZK3O402wf1c | and each one would be a vector
in nine-dimensional space |
ZK3O402wf1c | and we would be looking at
their linear combinations. |
ZK3O402wf1c | So we would be having
the linear combinations |
ZK3O402wf1c | of nine vectors in
nine-dimensional space, |
ZK3O402wf1c | and we would be trying to
find the combination that hit |
ZK3O402wf1c | the correct right-hand side b. |
ZK3O402wf1c | And we might also ask the
question can we always do it? |
ZK3O402wf1c | Can we get every
right-hand side b? |
ZK3O402wf1c | And certainly it will depend
on those nine columns. |
ZK3O402wf1c | Sometimes the answer
will be yes -- |
ZK3O402wf1c | if I picked a random matrix,
it would be yes, actually. |
ZK3O402wf1c | If I used MatLab and just used
the random command, picked |
ZK3O402wf1c | out a nine by nine matrix,
I guarantee it would be |
ZK3O402wf1c | good. |
ZK3O402wf1c | It would be
non-singular, it would |
ZK3O402wf1c | be invertible, all beautiful. |
ZK3O402wf1c | But if I choose those columns
so that they're not independent, |
ZK3O402wf1c | so that the ninth column is
the same as the eighth column, |
ZK3O402wf1c | then it contributes
nothing new and there |
ZK3O402wf1c | would be right-hand sides
b that I couldn't get. |
ZK3O402wf1c | Can you sort of think
about nine vectors |
ZK3O402wf1c | in nine-dimensional space
an take their combinations? |
ZK3O402wf1c | That's really the
central thought -- |
ZK3O402wf1c | that you get kind of used
to in linear algebra. |
ZK3O402wf1c | Even though you can't
really visualize it, |
ZK3O402wf1c | you sort of think you
can after a while. |
ZK3O402wf1c | Those nine columns and
all their combinations |
ZK3O402wf1c | may very well fill out the
whole nine-dimensional space. |
ZK3O402wf1c | But if the ninth column happened
to be the same as the eighth |
ZK3O402wf1c | column and gave nothing new,
then probably what it would |
ZK3O402wf1c | fill out would be -- |
ZK3O402wf1c | I hesitate even to say this --
it would be a sort of a plane |
ZK3O402wf1c | -- |
ZK3O402wf1c | an eight dimensional plane
inside nine-dimensional space. |
ZK3O402wf1c | And it's those eight
dimensional planes |
ZK3O402wf1c | inside nine-dimensional
space that we |
ZK3O402wf1c | have to work with eventually. |
ZK3O402wf1c | For now, let's stay with a nice
case where the matrices work, |
ZK3O402wf1c | we can get every
right-hand side b and here |
ZK3O402wf1c | we see how to do
it with columns. |
ZK3O402wf1c | Okay. |
ZK3O402wf1c | There was one step
which I realized |
ZK3O402wf1c | I was saying in words that I
now want to write in letters. |
ZK3O402wf1c | Because I'm coming back to the
matrix form of the equation, |
ZK3O402wf1c | so let me write it here. |
ZK3O402wf1c | The matrix form of my
equation, of my system |
ZK3O402wf1c | is some matrix A
times some vector x |
ZK3O402wf1c | equals some right-hand side b. |
ZK3O402wf1c | Okay. |
ZK3O402wf1c | So this is a multiplication. |
ZK3O402wf1c | A times x. |
ZK3O402wf1c | Matrix times vector,
and I just want to say |
ZK3O402wf1c | how do you multiply
a matrix by a vector? |
ZK3O402wf1c | Okay, so I'm just going
to create a matrix -- |
ZK3O402wf1c | let me take two
five one three -- |
ZK3O402wf1c | and let me take a vector
x to be, say, 1and 2. |
ZK3O402wf1c | How do I multiply a
matrix by a vector? |
ZK3O402wf1c | But just think a little
bit about matrix notation |
ZK3O402wf1c | and how to do that
in multiplication. |
ZK3O402wf1c | So let me say how I multiply
a matrix by a vector. |
ZK3O402wf1c | Actually, there are
two ways to do it. |
ZK3O402wf1c | Let me tell you my favorite way. |
ZK3O402wf1c | It's columns again. |
ZK3O402wf1c | It's a column at a time. |
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