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  • Okay.

  • This is lecture five in linear algebra.

  • And, it will complete this chapter of the book.

  • So the last section of this chapter

  • is two point seven that talks about permutations, which

  • finished the previous lecture, and transposes,

  • which also came in the previous lecture.

  • There's a little more to do with those guys, permutations

  • and transposes.

  • But then the heart of the lecture will be the beginning

  • of what you could say is the beginning of linear algebra,

  • the beginning of real linear algebra which is seeing

  • a bigger picture with vector spaces -- not just vectors,

  • but spaces of vectors and sub-spaces of those spaces.

  • So we're a little ahead of the syllabus, which

  • is good, because we're coming to the place

  • where, there's a lot to

  • do.

  • Okay.

  • So, to begin with permutations.

  • Can I just --

  • so these permutations, those are matrices P and they execute

  • row exchanges.

  • And we may need them.

  • We may have a perfectly good matrix,

  • a perfect matrix A that's invertible that we can solve A

  • x=b, but to do it --

  • I've got to allow myself that extra freedom

  • that if a zero shows up in the pivot position I move it away.

  • I get a non-zero.

  • I get a proper pivot there by exchanging from a row below.

  • And you've seen that already, and I just

  • want to collect the ideas together.

  • And principle, I could even have to do

  • that two times, or more times.

  • So I have to allow --

  • to complete the --

  • the theory, the possibility that I take my matrix A,

  • I start elimination, I find out that I need row exchanges

  • and I do it and continue and I finish.

  • Okay.

  • Then all I want to do is say -- and I won't make a big project

  • out of this --

  • what happens to A equal L U?

  • So A equal L U --

  • this was a matrix L with ones on the diagonal and zeroes

  • above and multipliers below, and this U

  • we know, with zeroes down here.

  • That's only possible.

  • That description of elimination assumes

  • that we don't have a P, that we don't have any row exchanges.

  • And now I just want to say, okay, how

  • do I account for row exchanges?

  • Because that doesn't.

  • The P in this factorization is the identity matrix.

  • The rows were in a good order, we left them there.

  • Maybe I'll just add a little moment of reality,

  • too, about how Matlab actually does elimination.

  • Matlab not only checks whether that pivot is not zero,

  • as every human would do.

  • It checks for is that pivot big enough,

  • because it doesn't like very, very small pivots.

  • Pivots close to zero are numerically bad.

  • So actually if we ask Matlab to solve a system,

  • it will do some elimination some row exchanges, which

  • we don't think are necessary.

  • Algebra doesn't say they're necessary, but accuracy --

  • numerical accuracy says they are.

  • Well, we're doing algebra, so here we

  • will say, well, what do row exchanges do,

  • but we won't do them unless we have to.

  • But we may have to.

  • And then, the result is --

  • it's hiding here.

  • It's the main fact.

  • This is the description of elimination with row exchanges.

  • So A equal L U becomes P A equal L U.

  • So this P is the matrix that does the row exchanges,

  • and actually it does them --

  • it gets the rows into the right order,

  • into the good order where pivots will not --

  • where zeroes won't appear in the pivot position,

  • where L and U will come out right as up here.

  • So, that's the point.

  • Actually, I don't want to labor that point,

  • that a permutation matrix --

  • and you remember what those were.

  • I'll remind you from last time of what the main points about

  • permutation matrices were --

  • and then just leave this factorization

  • as the general case.

  • This is -- any invertible A we get this.

  • For almost every one, we don't need a P.

  • But there's that handful that do need row exchanges,

  • and if we do need them, there they are.

  • Okay, finally, just to remember what P was.

  • So permutations, P is the identity matrix

  • with reordered rows.

  • I include in reordering the possibility that you just

  • leave them the same.

  • So the identity matrix is -- okay.

  • That's, like, your basic permutation matrix --

  • your do-nothing permutation matrix is the identity.

  • And then there are the ones that exchange two rows and then

  • the ones that exchange three rows and then then ones that

  • exchange four --

  • well, it gets a little --

  • it gets more interesting algebraically

  • if you've got four rows, you might exchange them

  • all in one big cycle.

  • One to two, two to three, three to four, four to one.

  • Or you might have -- exchange one and two and three and four.

  • Lots of possibilities there.

  • In fact, how many possibilities?

  • The answer was (n)factorial.

  • This is n(n-1)(n-2)...

  • (3)(2)(1).

  • That's the number of -- this counts the reorderings,

  • the possible reorderings.

  • So it counts all the n by n permutations.

  • And all those matrices have these --

  • have this nice property that they're all invertible,

  • because we can bring those rows back into the normal order.

  • And the matrix that does that is just P --

  • is just the same as the transpose.

  • You might take a permutation matrix,

  • multiply by its transpose and you will see how --

  • that the ones hit the ones and give the ones in the identity

  • matrix.

  • So this is a --

  • we'll be highly interested in matrices

  • that have nice properties.

  • And one property that -- maybe I could rewrite that as P

  • transpose P is the identity.

  • That tells me in other words that this

  • is the inverse of that.

  • Okay.

  • We'll be interested in matrices that have P transpose

  • P equal the identity.

  • There are more of them than just permutations,

  • but my point right now is that permutations are like a little

  • group in the middle --

  • in the center of these special matrices.

  • Okay.

  • So now we know how many there are.

  • Twenty four in the case of -- there are twenty four four

  • by four permutations, there are five factorial which is

  • a hundred and twenty, five times twenty four would bump us up

  • to a hundred and twenty -- so listing all the five by five

  • permutations would be not so much fun.

  • Okay.

  • So that's permutations.

  • Now also in section two seven is some discussion of transposes.

  • And can I just complete that discussion.

  • First of all, I haven't even transposed a matrix

  • on the board here, have I?

  • So I'd better do it.

  • So suppose I take a matrix like (1 2 4; 3 3 1).

  • It's a rectangular matrix, three by two.

  • And I want to transpose it.

  • So what's --

  • I'll use a T, also Matlab would use a prime.

  • And the result will be --

  • I'll right it here, because this was three rows and two columns,

  • this was a three by two matrix.

  • The transpose will be two rows and three columns,

  • two by three.

  • So it's short and wider.

  • And, of course, that row -- that column becomes a row --

  • that column becomes the other row.

  • And at the same time, that row became a column.

  • This row became a column.

  • Oh, what's the general formula for the transpose?

  • So the transpose --

  • you see it in numbers.

  • What I'm going to write is the same thing in symbols.

  • The numbers are the clearest, of course.

  • But in symbols, if I take A transpose

  • and I ask what number is in row I and column J of A transpose?

  • Well, it came out of A.

  • It came out A by this flip across the main diagonal.

  • And, actually, it was the number in A

  • which was in row J, column I.

  • So the row and column --

  • the row and column numbers just get reversed.

  • The row number becomes the column number,

  • the column number becomes the row number.

  • No problem.

  • Okay.

  • Now, a special --

  • the best matrices, we could say.

  • In a lot of applications, symmetric matrices show up.

  • So can I just call attention to symmetric matrices?

  • What does that mean?

  • What does that word symmetric mean?

  • It means that this transposing doesn't change the matrix.

  • A transpose equals A.

  • And an example.

  • So, let's take a matrix that's symmetric,

  • so whatever is sitting on the diagonal --

  • but now what's above the diagonal, like a one,

  • had better be there, a seven had better be here,

  • a nine had better be there.

  • There's a symmetric matrix.

  • I happened to use all positive numbers as its entries.

  • That's not the point.

  • The point is that if I transpose that matrix,

  • I get it back again.

  • So symmetric matrices have this property A transpose equals A.

  • I guess at this point --

  • I'm just asking you to notice this family of matrices that

  • are unchanged by transposing.

  • And they're easy to identify, of course.

  • You know, it's not maybe so easy before we had a case where

  • the transpose gave the inverse.

  • That's highly important, but not so simple to see.

  • This is the case where the transpose gives the same matrix

  • back again.

  • That's totally simple to see.

  • Okay.

  • Could actually -- maybe I could even say when would we get such

  • a matrix?

  • For example, this -- that matrix is absolutely far from

  • symmetric, right?

  • The transpose isn't even the same shape --

  • because it's rectangular, it turns the --

  • lies down on its side.

  • But let me tell you a way to get a symmetric matrix out of

  • this.

  • Multiply those together.

  • If I multiply this rectangular, shall I

  • call it R for rectangular?

  • So let that be R for rectangular matrix

  • and let that be R transpose, which it is.

  • Then I think that if I multiply those together,

  • I get a symmetric matrix.

  • Can I just do it with the numbers

  • and then ask you why, how did I know it would be symmetric?

  • So my point is that R transpose R is always symmetric.

  • Okay?

  • And I'm going to do it for that particular R transpose R which

  • was --

  • let's see, the column was one two four three three one.

  • I called that one R transpose, didn't I,

  • and I called this guy one two four three three one.

  • I called that R.

  • Shall we just do that multiplication?

  • Okay, so up here I'm getting a ten.

  • Next to it I'm getting two, a nine, I'm getting an eleven.

  • Next to that I'm getting four and three, a seven.

  • Now what do I get there?

  • This eleven came from one three times two three, right?

  • Row one, column two.

  • What goes here?

  • Row two, column one.

  • But no difference.

  • One three two three or two three one three, same thing.

  • It's going to be an eleven.

  • That's the symmetry.

  • I can continue to fill it out.

  • What -- oh, let's get that seven.

  • That seven will show up down here, too,

  • and then four more numbers.

  • That seven will show up here because one three times four

  • one gave the seven, but also four one times one three

  • will give that seven.

  • Do you see that it works?

  • Actually, do you want to see it work also in matrix language?

  • I mean, that's quite convincing, right?

  • That seven is no accident.

  • The eleven is no accident.

  • But just tell me how do I know if I transpose this guy --

  • How do I know it's symmetric?

  • Well, I'm going to transpose it.

  • And when I transpose it, I'm hoping

  • I get the matrix back again.

  • So can I transpose R transpose R?

  • So just -- so, why?

  • Well, my suggestion is take the transpose.

  • That's the only way to show it's symmetric.

  • Take the transpose and see that it didn't change.

  • Okay, so I take the transpose of R transpose R.

  • Okay.

  • How do I do that?

  • This is our little practice on the rules for transposes.

  • So the rule for transposes is the order gets reversed.

  • Just like inverses, which we did prove,

  • same rule for transposes and -- which we'll now use.

  • So the order gets reversed.

  • It's the transpose of that that comes first,

  • and the transpose of this that comes -- no.

  • Is that -- yeah.

  • That's what I have to write, right?

  • This is a product of two matrices and I want its

  • transpose.

  • So I put the matrices in the opposite order

  • and I transpose them.

  • But what have I got here?

  • What is R transpose transpose?

  • Well, don't all speak at once.

  • R transpose transpose, I flipped over the diagonal,

  • I flipped over the diagonal again, so I've got R.

  • And that's just my point, that if I started with this matrix,

  • I transposed it, I got it back again.

  • So that's the check, without using numbers, but with --

  • it checked in two lines that I always get symmetric matrices

  • this way.

  • And actually, that's where they come

  • from in so many practical applications.

  • Okay.

  • So now I've said something today about permutations and about

  • transposes and about symmetry and I'm ready

  • for chapter three.

  • Can we take a breath --

  • the tape won't take a breath, but the lecturer will,

  • because to tell you about vector spaces is --

  • we really have to start now and think, okay, listen up.

  • What are vector spaces?

  • And what are sub-spaces?

  • Okay.

  • So, the point is, The main operations that we do --

  • what do we do with vectors?

  • We add them.

  • We know how to add two vectors.

  • We multiply them by numbers, usually called scalers.

  • If we have a vector, we know what three V is.

  • If we have a vector V and W, we know what V plus W is.

  • Those are the two operations that we've

  • got to be able to do.

  • To legitimately talk about a space of vectors,

  • the requirement is that we should

  • be able to add the things and multiply by numbers

  • and that there should be some decent rules satisfied.

  • Okay.

  • So let me start with examples.

  • So I'm talking now about vector spaces.

  • And I'm going to start with examples.

  • Let me say again what this word space is meaning.

  • When I say that word space, that means to me

  • that I've got a bunch of vectors, a space of vectors.

  • But not just any bunch of vectors.

  • It has to be a space of vectors --

  • has to allow me to do the operations that vectors

  • are for.

  • I have to be able to add vectors and multiply by numbers.

  • I have to be able to take linear combinations.

  • Well, where did we meet linear combinations?

  • We met them back in, say in R^2.

  • So there's a vector space.

  • What's that vector space?

  • So R two is telling me I'm talking about real numbers

  • and I'm talking about two real numbers.

  • So this is all two dimensional vectors --

  • real, such as --

  • well, I'm not going to be able to list them all.

  • But let me put a few down.

  • |3; 2|, |0;0|, |pi; e|.

  • So on.

  • And it's natural -- okay.

  • Let's see, I guess I should do algebra first.

  • Algebra means what can I do to these vectors?

  • I can add them.

  • I can add that to that.

  • And how do I do it?

  • A component at a time, of course.

  • Three two added to zero zero gives me, three two.

  • Sorry about that.

  • Three two added to pi e gives me three plus pi, two plus e.

  • Oh, you know what it does.

  • And you know the picture that goes with it.

  • There's the vector three two.

  • And often, the picture has an arrow.

  • The vector zero zero, which is a highly important vector --

  • it's got, like, the most important here

  • -- is there.

  • And of course there's not much of an arrow.

  • Pi -- I'll have to remember -- pi is about three and a little

  • more, e is about two and a little more.

  • So maybe there's pi e.

  • I never drew pi e before.

  • It's just natural to --

  • this is the first component on the horizontal

  • and this is the second component,

  • going up the vertical.

  • Okay.

  • And the whole plane is R two.

  • So R two is, we could say, the plane.

  • The xy plane.

  • That's what everybody thinks.

  • But the point is it's a vector space because all those vectors

  • are in there.

  • If I removed one of them --

  • Suppose I removed zero zero.

  • Suppose I tried to take the -- considered the X Y plane with

  • a puncture, with a point removed.

  • Like the origin.

  • That would be, like, awful to take the origin away.

  • Why is that?

  • Why do I need the origin there?

  • Because I have to be allowed -- if I had these other vectors,

  • I have to be allowed to multiply three two --

  • this was three two --

  • by anything, by any scaler, including zero.

  • I've got to be allowed to multiply by zero

  • and the result's got to be there.

  • I can't do without that point.

  • And I have to be able to add three two to the opposite guy,

  • minus three minus two.

  • And if I add those I'm back to the origin again.

  • No way I can do without the origin.

  • Every vector space has got that zero vector in it.

  • Okay, that's an easy vector space,

  • because we have a natural picture of it.

  • Okay.

  • Similarly easy is R^3.

  • This would be all -- let me go up a little here.

  • This would be --

  • R three would be all three dimensional vectors --

  • or shall I say vectors with three real components.

  • Okay.

  • Let me just to be sure we're together,

  • let me take the vector three two zero.

  • Is that a vector in R^2 or R^3?

  • Definitely it's in R^3.

  • It's got three components.

  • One of them happens to be zero, but that's a perfectly okay

  • number.

  • So that's a vector in R^3.

  • We don't want to mix up the --

  • I mean, keep these vectors straight and keep R^n straight.

  • So what's R^n?

  • R^n.

  • So this is our big example, is all vectors with n components.

  • And I'm making these darn things column vectors.

  • Can I try to follow that convention,

  • that they'll be column vectors, and their components should

  • be real numbers.

  • Later we'll need complex numbers and complex vectors,

  • but much later.

  • Okay.

  • So that's a vector space.

  • Now, let's see.

  • What do I have to tell you about vector spaces?

  • I said the most important thing, which is that we can add any

  • two of these and we -- still in R^2.

  • We can multiply by any number and we're still in R^2.

  • We can take any combination and we're still in R^2.

  • And same goes for R^n.

  • It's -- honesty requires me to mention that these operations

  • of adding and multiplying have to obey a few rules.

  • Like, we can't just arbitrarily say, okay, the sum of three two

  • and pi e is zero zero.

  • It's not.

  • The sum of three two and minus three two is zero zero.

  • So -- oh, I'm not going to -- the book, actually,

  • lists the eight rules that the addition and multiplication

  • have to satisfy, but they do.

  • They certainly satisfy it in R^n and usually it's not those

  • eight rules that are in doubt.

  • What's -- the question is, can we do those additions and do we

  • stay in the space?

  • Let me show you a case where you can't.

  • So suppose this is going to be not a vector space.

  • Suppose I take the xy plane -- so there's R^2.

  • That is a vector space.

  • Now suppose I just take part of it.

  • Just this.

  • Just this one -- this is one quarter of the vector space.

  • All the vectors with positive or at least not negative

  • components.

  • Can I add those safely?

  • Yes.

  • If I add a vector with, like, two --

  • three two to another vector like five six,

  • I'm still up in this quarter, no problem with adding.

  • But there's a heck of a problem with multiplying by scalers,

  • because there's a lot of scalers that will take me out

  • of this quarter plane, like negative ones.

  • If I took three two and I multiplied by minus five,

  • I'm way down here.

  • So that's not a vector space, because it's not --

  • closed is the right word.

  • It's not closed under multiplication

  • by all real numbers.

  • So a vector space has to be closed under multiplication

  • and addition of vectors.

  • In other words, linear combinations.

  • It -- so, it means that if I give you a few vectors --

  • yeah look, here's an important -- here --

  • now we're getting to some really important vector spaces.

  • Well, R^n -- like, they are the most important.

  • But we will be interested in so- in vector spaces that are

  • inside R^n.

  • Vector spaces that follow the rules, but they --

  • we don't need all of -- see, there we started with R^2 here,

  • and took part of it and messed it up.

  • What we got was not a vector space.

  • Now tell me a vector space that is part of R^2 and is still

  • safely -- we can multiply, we can add and we stay in this

  • smaller vector space.

  • So it's going to be called a subspace.

  • So I'm going to change this bad example to a good one.

  • Okay.

  • So I'm going to start again with R^2,

  • but I'm going to take an example -- it is a vector space,

  • so it'll be a vector space inside R^2.

  • And we'll call that a subspace of R^2.

  • Okay.

  • What can I do?

  • It's got something in it.

  • Suppose it's got this vector in it.

  • Okay.

  • If that vector's in my little subspace

  • and it's a true subspace, then there's

  • got to be some more in it,

  • right?

  • I have to be able to multiply that by two,

  • and that double vector has to be included.

  • Have to be able to multiply by zero, that vector,

  • or by half, or by three quarters.

  • All these vectors.

  • Or by minus a half, or by minus one.

  • I have to be able to multiply by any number.

  • So that is going to say that I have to have that whole line.

  • Do you see that?

  • Once I get a vector in there --

  • I've got the whole line of all multiples of that vector.

  • I can't have a vector space without extending to get

  • those multiples in there.

  • Now I still have to check addition.

  • But that comes out okay.

  • This line is going to work, because I could add something

  • on the line to something else on the line

  • and I'm still on the line.

  • So, example.

  • So this is all examples of a subspace --

  • our example is a line in R^2 actually -- not just any line.

  • If I took this line, would that --

  • so all the vectors on that line.

  • So that vector and that vector and this vector and this vector

  • --

  • in lighter type, I'm drawing something that doesn't work.

  • It's not a subspace.

  • The line in R^2 -- to be a subspace,

  • the line in R^2 must go through the zero vector.

  • Because -- why is this line no good?

  • Let me do a dashed line.

  • Because if I multiplied that vector on the dashed line

  • by zero, then I'm down here, I'm not on the dashed line.

  • Z- zero's got to be.

  • Every subspace has got to contain zero --

  • because I must be allowed to multiply by zero and that will

  • always give me the zero vector.

  • Okay.

  • Now, I was going to make --

  • create some subspaces.

  • Oh, while I'm in R^2, why don't we think of all

  • the possibilities.

  • R two, there can't be that many.

  • So what are the possible subspaces of R^2?

  • Let me list them.

  • So I'm listing now the subspaces of R^2.

  • And one possibility that we always allow

  • is all of R two, the whole thing, the whole space.

  • That counts as a subspace of itself.

  • You always want to allow that.

  • Then the others are lines --

  • any line, meaning infinitely far in both directions

  • through the zero.

  • So that's like the whole space --

  • that's like whole two D space.

  • This is like one dimension.

  • Is this line the same as R^1 ?

  • No.

  • You could say it looks a lot like R^1.

  • R^1 was just a line and this is a line.

  • But this is a line inside R^2.

  • The vectors here have two components.

  • So that's not the same as R^1, because there the vectors only

  • have one component.

  • Very close, you could say, but not the same.

  • Okay.

  • And now there's a third possibility.

  • There's a third subspace that's --

  • of R^2 that's not the whole thing, and it's not a line.

  • It's even less.

  • It's just the zero vector alone.

  • The zero vector alone, only.

  • I'll often call this subspace Z, just for zero.

  • Here's a line, L.

  • Here's a plane, all of R^2.

  • So, do you see that the zero vector's okay?

  • You would just -- to understand subspaces,

  • we have to know the rules -- and knowing the rules means that we

  • have to see that yes, the zero vector by itself,

  • just this guy alone satisfies the rules.

  • Why's that?

  • Oh, it's too dumb to tell you.

  • If I took that and added it to itself, I'm still there.

  • If I took that and multiplied by seventeen, I'm still there.

  • So I've done the operations, adding and multiplying

  • by numbers, that are required, and I didn't go

  • outside this one point space.

  • So that's always -- that's the littlest subspace.

  • And the largest subspace is the whole thing and in-between come

  • all --

  • whatever's in between.

  • Okay.

  • So for example, what's in between for R^3?

  • So if I'm in ordinary three dimensions, the subspace is R,

  • all of R^3 at one extreme, the zero vector at the bottom.

  • And then a plane, a plane through the origin.

  • Or a line, a line through the origin.

  • So with R^3, the subspaces were R^3, plane through the origin,

  • line through the origin and a zero vector by itself,

  • zero zero zero, just that single vector.

  • Okay, you've got the idea.

  • But, now comes --

  • the reality is --

  • what are these -- where do these subspaces come --

  • how do they come out of matrices?

  • And I want to take this matrix --

  • oh, let me take that matrix.

  • So I want to create some subspaces out of that matrix.

  • Well, one subspace is from the columns.

  • Okay.

  • So this is the important subspace,

  • the first important subspace that comes from that matrix --

  • I'm going to -- let me call it A again.

  • Back to -- okay.

  • I'm looking at the columns of A.

  • Those are vectors in R^3.

  • So the columns are in R^3.

  • The columns are in R^3.

  • So I want those columns to be in my subspace.

  • Now I can't just put two columns in my subspace

  • and call it a subspace.

  • What do I have to throw in -- if I'm going to put those two

  • columns in, what else has got to be there to have a subspace?

  • I must be able to add those things.

  • So the sum of those columns --

  • so these columns are in R^3, and I have to be able --

  • I'm, you know, I want that to be in my subspace,

  • I want that to be in my subspace,

  • but therefore I have to be able to multiply them by anything.

  • Zero zero zero has got to be in my subspace.

  • I have to be able to add them so that four five five

  • is in the subspace.

  • I've got to be able to add one of these plus three of these.

  • That'll give me some other vector.

  • I have to be able to take all the linear combinations.

  • So these are columns in R^3 and all there linear combinations

  • form a subspace.

  • What do I mean by linear combinations?

  • I mean multiply that by something,

  • multiply that by something and add.

  • The two operations of linear algebra, multiplying by numbers

  • and adding vectors.

  • And, if I include all the results,

  • then I'm guaranteed to have a subspace.

  • I've done the job.

  • And we'll give it a name --

  • the column space.

  • Column space.

  • And maybe I'll call it C of A.

  • C for column space.

  • There's an idea there that --

  • Like, the central idea for today's lecture is --

  • got a few vectors.

  • Not satisfied with a few vectors,

  • we want a space of vectors.

  • The vectors, they're in -- these vectors in -- are in R^3 ,

  • so our space of vectors will be vectors in R^3.

  • The key idea's -- we have to be able to take

  • their combinations.

  • So tell me, geometrically, if I drew all these things --

  • like if I drew one two four, that would be somewhere maybe

  • there.

  • If I drew three three one, who knows, might be --

  • I don't know, I'll say there.

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

  • What else -- what's in the whole column space?

  • How do I draw the whole column space now?

  • I take all combinations of those two vectors.

  • Do I get -- well, I guess I actually listed

  • the possibilities.

  • Do I get the whole space?

  • Do I get a plane?

  • I get more than a line, that's for sure.

  • And I certainly get more than the zero vector,

  • but I do get the zero vector included.

  • What do I get if I combine --

  • take all the combinations of two vectors in R^3 ?

  • So I've got all this stuff on --

  • that whole line gets filled out, that whole line gets filled

  • out, but all in-between gets filled out --

  • between the two lines because I --

  • I allowed to add something from one line, something

  • from the other.

  • You see what's coming?

  • I'm getting a plane.

  • That's my -- and it's through the origin.

  • Those two vectors, namely one two four and three three one,

  • when I take all their combinations,

  • I fill out a whole plane.

  • Please think about that.

  • That's the picture you have to see.

  • You sure have to see it in R^3 , because we're going to do it

  • in R^10, and we may take a combination of five vectors

  • in R^10, and what will we have?

  • God knows.

  • It's some subspace.

  • We'll have five vectors.

  • They'll all have ten components.

  • We take their combinations.

  • We don't have R^5 , because our vectors have ten components.

  • And we possibly have, like, some five dimensional flat thing

  • going through the origin for sure.

  • Well, of course, if those five vectors were all on the line,

  • then we would only get that line.

  • So, you see, there are, like, other possibilities here.

  • It depends what -- it depends on those five vectors.

  • Just like if our two columns had been on the same line,

  • then the column space would have been only a line.

  • Here it was a plane.

  • Okay.

  • I'm going to stop at that point.

  • That's the central idea of -- the great example of how

  • to create a subspace from a matrix.

  • Take its columns, take their combinations,

  • all their linear combinations and you get the column space.

  • And that's the central sort of --

  • we're looking at linear algebra at a higher level.

  • When I look at A -- now, I want to look at Ax=b.

  • That'll be the first thing in the next lecture.

  • How do I understand Ax=b in this language --

  • in this new language of vector spaces and column spaces.

  • And what are other subspaces?

  • So the column space is a big one, there are others to come.

  • Okay, thanks.

Okay.

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