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[0:10]OK, this is linear algebra, lecture 9. And this is a key lecture, this is where we get these ideas of linear independence, when a bunch of vectors are independent; or dependent, that’s the opposite.
[0:33]The space they span, a basis for a subspace, or a basis for a vector space, that’s the central idea, and then the dimension of that subspace. So, this is the day that those words gets assign a clearer meaning. And emphasize that we talk about a bunch of vectors being independent – wouldn’t talk about a matrix being independent – a bunch of vectors being independent, a bunch of vectors spanning a space, a bunch of vectors being a basis. And the dimension is some number.
[1:14]OK. So what are the definitions? Can I begin with a fact, a highly important fact that… I didn’t call directly attention to earlier. Suppose I have a matrix, and I looked at Ax = 0. Suppose the matrix has a lot of columns. So that n is bigger than m. So, I’m looking at n equations… I’m sorry, m equations, the small number of equations, m, and more unknowns. I have more unknowns than equations.
[1:57]Let me write that down, more unknowns than equations. MORE UNKNOWN x’s than equations. Then the conclusion is that there’s something in the nullspace of A, other than just a 0-vector.
[2:21]The conclusion is there are some nonzero x’s, solves that Ax = 0.There are some special solutions, and why? We know why, I mean it’s sort of like seems like a reasonable thing, more unknowns than equation, then it’s seems reasonable that we can solve them.
[2:45]But we have a clear algorithm, which starts with this system, and does elimination; get the thing in echelon form, with some pivots and pivot columns, and possibly some free columns that don’t have pivots. And the point is here there will be some free columns.
[3:10]The reason, so the reason is THERE WILL BE FREE VARIABLES, at least 1 – that’s the reason. We now have a complete algorithm, a complete systematic way to say, OK, we take this system of Ax = 0, we row reduced, we identify the free variables and since there’re n variables, and at most m pivots, there will be some free variables – at least one, at least n – m in fact, left over, and those are variables I can assign nonzero values to.
[4:00]I don’t have to set those to 0; I can take them to be 1, or whatever I like, and then I can solve for the pivot variables, so then it gives me a solution, to Ax = 0, and it’s the solution that isn’t all 0’s.
[4:15]So, uh… that’s a crucial… That’s an important point that we’ll use now in this lecture. So now I want to say what does it mean for a bunch of vectors to be independent. OK. So this is like the background that we know. Now I want to speak about INDEPENDENCE. OK. Let’s see. I can give you the abstract definition, and I will, but I would also like to give you the direct meaning when my… So the question is when VECTORS x1, x2… up to… suppose I have n vectors, ARE INDEPENDENT, IF… Now I have to give you... or linearly independent… I’ll often just to say and write independent for short.
[5:32]If…OK. Let… I’ll give you the full definition. These are just vectors in some vector space. I can take combinations of them. The question is: do any combination gives 0? If some combination of those vectors gives a 0-vector other than the combination of all 0’s, then they’re dependent. They are independent if NO COMBINATION GIVES THE 0 VECTOR, and I have to put in an EXCEPT, THE 0 COMBINATION.
[6:19]So what do I mean by that? No combination gives the 0 vector, any combination c1x2 + c2x2 + … + cnxn is not 0! Except for the 0 combination this is when ALL THE c’s ARE 0. Then of course, that combination I know I’ll get 0. But the question is does any other combination gives 0. If not, then the vectors are independent. If some of the combination does give 0, the vectors are dependent.
[7:05]OK, let’s just take examples. Suppose I’m in, say in 2-dimensional space. OK. I give you… I’d like to first take an example, let me take an example where I have a vector, and twice that vector. So that’s 2 vectors, v and 2v. Are those dependent or independent?
[7:35]Those are dependent for sure. One vector is twice the other; one vector is twice as long as the other. So the word dependent means anything. This should be dependent, and they are. And in fact, I would take 2 of the first… So here’s the vector v, and the other guy is the vector 2v. That’s my… So there’s the vector v1, and my next vector v2 is 2v1. Of course those are dependent, because 2 of this first vectors minus the second vector is 0. That’s a combination of these 2 vectors that gives the 0-vector.
[8:19]OK, that was clear! Suppose I have a vector… here’s another example – it’s an easy example. Suppose I have a vector, and the other guy is a 0-vector. Suppose I have a vector v1, and v2 is the 0-vector then are those vectors dependent or independent? They are dependent again! You could say, well this guy is 0 times that one. This one is some combination of those.
[8:53]But let me write it the other way, let me say what combination, how many v1’s and how many v2’s, so I take, to get the 0 vector? If v1 is like the vector (2, 1), and v2 is the 0 vector (0, 0), then I would like to show that some combination of those gives the 0 vector. What shall I take? How many v1 shall I take? 0 of them! Yeah, I take no v1’s, but how many v2’s? Say a 6! OK, or 5.
[9:35]Then, in other words, the point is, if the 0 vector is in there, if one of these vectors is the 0 vector, independence is dead, right? If one of the vectors is the 0 vector, then I could always take, include that one and none of the others, and I would get the 0 answer, and I would show dependence.
[10:00]OK, now let me finally draw an example, where they will be independent. Suppose that’s v1, and that’s v2. those are surely independent, right? Any combination of v1 and v2 will not be 0, except the 0 combination. So those would be independent. But now let me stick in a third vector, v3, independent or dependent now, those 3 vectors?
[10:35]So, now, n is 3 here. I’m in 2-dimensional space, whatever I’m in the plane, I have 3 vectors, that I didn’t draw so carefully, I didn’t even tell what exactly they were, but what’s this answer on dependent or independent? Dependent! Why, how do I know that those are dependent?
[11:02]How do I know that some combination of v1, v2 and v3 gives me the 0 vector? I know because of that. That’s the key fact that tells me that 3 vectors in a plane have to be dependent. Why’s that? What’s the connection between the dependence of these 3 vectors and that fact?
[11:33]OK. So, here’s the connection. I take the matrix A that has v1 in its first column, v2 in its second column, v3 in its third column. So it got 3 columns and v1… I don’t know… that looks like about (2, 1) to be; v2 looks like it might be (1, 2), v3 looks like it might be, maybe (2.5, -1).
[12:08]OK. Those are my 3 vectors and I put them in the columns of A. Now that matrix A is 2 by 3. It’s just this pattern that where we know we’ve got extra variables; we know we have some free variables; we know that there’re some combination and let me, instead of x’s, let me call them c1, c2, and c3, that gives the 0-vector.
[12:41]Sorry that my little bit of art got in the way. Do you see the point? When I have a matrix, I’m interested in whether its columns are dependent or independent. Their columns are dependent if there is something in the nullspace. The columns are dependent because this thing in the nullspace says that c1 of that plus c2 of that, plus c3 of this is 0.
[13:10]So, in other words, I can go out some v1, out some more v2, back on v3, and end up 0. OK. Here I’ve given the general abstract definition, but let me repeat that definition. So, this is like a repeat. WHEN the v1 – let me call them v’s now – v1 up to vn ARE THE COLUMNS OF A. In other words, this is telling me if I’m in m dimensional space, like 2-dimensional space in the example, I can answer that dependence/independence question directly by putting those vectors in the columns of a matrix.
[14:17]then THEY ARE INDEPENDENT, IF THE NULLSPACE OF A IS, is what? If I have a bunch of columns in a matrix, I’m looking at their combinations, but that’s just A times the vector of c’s. And these columns will be independent if the nullspace of A is only the 0 vector, is the 0 vector.
[15:04]THEY ARE DEPENDENT if there’s something else in there, if there’s something else in the nullspace, IF Ac = 0, FOR SOME NONZERO c in the nullspace. Then they’re dependent, because that’s telling me a combination of columns gives the 0-column.
[15:38]I think you’re with me, because we’ve seen, lecture after lecture, we’ve looking at the combinations of the columns and asking, do we get 0 or don’t we. And now we’re giving the official name: dependent if we do; independent if we don’t. So, I could express this in other words now, I could say the rank, what’s the rank in this independent case? The rank r of the matrix in the case of independent columns is…
[16:15]So, the columns are independent, so how many pivot columns have I got? All n. All the columns would be pivot columns. Because free columns, they’re telling me that there are combination of earlier columns. So this would be the case where the rank is n; this would be the case where the rank is smaller than n.
[16:43]So, in this case the rank is n, and the nullspace of A is only the 0-vector. And no free variables. NO FREE VARIABLES. And this is the case, YES FREE VARIABLES. If you allow me to stretch the English language that far… Uh… That’s the case where we have a combination that gives the 0 column.
[17:21]So, I’m often interested in the case when my vectors are popped into a matrix. So the definition over there of independence didn’t talk about any matrix, the vectors didn’t have to be vectors in n dimensional space. And I want to give you some examples of vectors that aren’t what you think of immediately as vectors.
[17:41]But most of times, this is the vectors we think of r columns, and we can put them in a matrix, and then independence or dependence comes back to the nullspace. OK, that’s the idea of independence.
[18:08]Can I just… Let me go on to spanning a space. What does that mean, for a bunch of vectors to span a space? Well actually we’ve seen it already. You remember if we had columns in a matrix, we took all their combinations and that gave us a column space. Those vectors that we started with, span that column space.
[18:44]So, spanning a space means… So let me move that important stuff right up… OK. So, vectors, let me call them, say v1, up to… I’ll use some different letters, say vl, span a subspace, or just a vector space, I could say, SPAN A SPACE MEANS THE SPACE CONSISTS OF ALL COMBINATIONS OF THOSE VECTORS.
[19:48]That’s exactly what we did with the column space. So now I could say in short hand, the columns of a matrix span the column space. So remember the bunch of vectors that has this property that they span a space. And actually, if I gave you a bunch of vectors and say, OK, let s be the space that they spanned, in other words, let s contain all their combinations. That’s space s will be the smallest space with those vectors in it, right?
[20:24]Because any space with those vectors in it must have all the combinations of those vectors in it. And if I stop there, then I’ll get the smallest space, and that’s the space that they spanned. OK, so I’m just rather than meaning to say, take all linear combinations and put them in a space, I’m compressing that into the word SPAN – straightforward! OK.
[20:59]So, if I think of the column space of the matrix, I’ve got the… So, I start with the columns, I take all their combinations that gives me the column space – they span a column space. Now, are they independent? Maybe yes maybe no. They depend on the particular columns that went into that matrix.
[21:27]But obviously I’m highly interested in a set of vectors that spans a space and is independent. That means that I’ve got the right number of vectors, if I didn’t have all of them, I wouldn’t have my whole space. If had more than that, they wouldn’t be independent. So, like basis, and that the word that’s coming, is just right.
[22:01]So here let me just put what that word’s mean. A BASIS FOR A VECTOR SPACE IS A SEQUENCE OF VECTORS, so I’ll call them v1, v2, up to, let me say, vd now, I’ll stop with that letter’s, that has 2 properties. I’ve got enough vectors, and not too many. It’s the natural idea of a basis, so a basis is a bunch of vectors in the space, and… So it’s a sequence of vectors, WITH 2 PROPERTIES. ONE, THEY ARE INDEPENDENT, AND TWO, you know what’s coming? THEY SPAN THE SPACE.
[23:23]OK. Let me take… So time for examples of course. So I’m asking you now to put definition 1, the definition of independence together with definition 2, and let’s look at the examples. Because this combination means the vectors I have is just right. And so this idea of basis will be central.
[23:55]I’ll always be asking you now for a basis, whenever I looked at the subspace. If you give me a basis for that subspace, you’ve told me what it is. You’ve told me everything I need to know about that subspace. I take their combinations and I know that I need all the combinations.
[24:15]OK. Examples! OK, so examples of the basis. Let me start with 2-dimensional space. Suppose the space, say example... the space is... Ah, let’s make it R3 – real 3-dimensional space. Give me one basis, ONE BASIS IS... So I want some vectors... If I ask you for a basis, I’m asking for vectors, a little list of vectors, and there should be just right.
[25:00]So what would be a basis for 3-dimensional space? Well, the first basis that comes to mind... What do we write that down? The first space that comes to mind is this vector, this vector, and this vector. OK. That’s one basis, not the only basis – that’s gonna be my point. But let’s just say, yes, that’s a basis.
[25:34]Are those vectors independent? So, that it likes the X, Y, and Z axis. So, if those are not independent, we are in trouble. Certainly, they are! Take a combination, c1 of this vector + c2 of this vector + c3 of that vector and try to make it give a 0-vector. What are the c’s?
[26:02]If c1 of that + c2 of that + c3 of that gives me (0, 0, 0), then the c’s are all 0, right! So, that’s the test for independence. In the language of matrices, which was under that board, I could make those the columns of a matrix, while it would be the identity matrix. Then I would ask: what’s the nullspace of the identity matrix? And you would say: it’s only the 0-vector. And I would say: fine, then the columns are independent.
[26:38]The only thing the identity times a vector giving 0 – the only vector that does that is the 0. OK. Now, that’s not the only basis far from it. Tell me another basis, a second basis, ANOTHER BASIS. So, give me... well, I’ll just start it out: (1, 1, 2); (2, 2, 5)... Suppose I stop here. Has that little bunch of vectors got the properties that I’m asking for, in a basis for R3? We are looking for a basis for R3. Are they independent, those 2 column vectors?
[27:33]Yes! Do they span R3? No, our feeling is no. Our feeling is that there are some vectors in R3 that are not combinations of those. OK, suppose I add in a... I need another vector then, because these 2 don’t span the space. OK, now it would be foolish for me to put in (3, 3, 7), right, as the third vector? That would be a goof. Because... if I put in (3, 3, 7), those vectors would be dependent, right? If I put in (3, 3, 7), it would be the sum of those 2; it would lie in the same plane as those; it wouldn’t be independent; I wouldn’t... my attempt to create a basis would be dead.
[28:25]But if I take... So what vector can I take? I can take any vector that’s not in that plane. Let me try... I hope the (3, 3, 8) would do it. At least it’s not the sum of those 2 vectors. I believe that’s a basis. And what’s the test then, for that to be a basis? Because I just pick those numbers, and if I picked (5, 7, -14), how would we know do we have a basis or don’t we?
[29:05]You would put them into the columns of a matrix, and you would do elimination, row reduction, and you would see, do you get any free variables, or are all the columns pivot columns. Well now actually the matrix would be 3 by 3, so what’s the test on the matrix then?
[29:31]The matrix... So, in this case, when my space is R3, and I have 3 vectors, my matrix is square, and what I’m asking is about that matrix in order for those columns to be a basis. So in this... For Rn, if I have n VECTORS GIVE A BASIS, IF THE n by n MATRIX, WITH THOSE COLUMNS IS what? What’s the requirement on that matrix? Invertible! Right, right. The matrix should be invertible, for a square matrix, that’s the perfect answer: IS INVERTIBLE.
[30:42]So, that’s when the space is the whole space Rn. Let me be sure you’re with me here. Let me remove that. Are those 2 vectors a basis for any space at all? Is there are vector space that those really are a basis for? That pair of vectors? This guy and this one, (1, 1, 2) and (2, 2, 5), is there’s a space for which that’s the basis?
[31:18]Sure! They’re independent, so they satisfy the first requirement. So what space should I take for them to be a basis of? What’s basis will they be a basis for? The one they span – their combinations. It’s a plane, right? It would be a plane inside R3. So, if I take this vector (1, 1, 2), say goes there, and this vector (2, 2, 5), say goes there. Those are a basis for... if they span a plane, and they’re basis of the plane because they’re independent. If I stick in some third guy, some (3, 3, 7), which is in the plane, suppose I try to put in (3, 3, 7), then the three vectors would still span the plane, but they wouldn’t be a basis anymore because they’re not independent anymore.
[32:14]OK. So, we’re looking at the question of.... So, again, the case with n independent columns is the case where the column vectors span the column space. They are independent, so they are the basis for the column space.
[32:43]Now, there’s one bit of intuition... Let me go back to all of Rn, so, where I put (3, 3, 8). OK, the first message is that the basis is not unique, right? There zillions of basis. I take any invertible 3 by 3 matrix, its columns are basis for R3. The column space is R3, and if those, if that matrix is invertible, those columns are independent. I’ve got a basis for R3.
[33:22]So, there are many many bases, but there is something in common, for all of those bases. There are something that this basis shares with that basis, and every other basis for R3. And what’s that?
[33:45]Well, you saw it coming because when I stop here, and ask if that was a basis for R3, you said no. And I know that you said no, because you knew there weren’t enough vectors there. And the great fact is that there are many many bases, but... Let me put in somebody else, just for variety.
[34:17]There are many many bases but they all have the same number of vectors. If we’re talking about this space R3, then that number of vectors is 3. If we’re talking about the space Rn, then that number of vectors is n.
[34:35]If we’re talking about some other space, the column space of some matrix, or the nullspace of some matrix, or some other space that we haven’t even thought of, then that still is true that every basis, that there are lots of bases, but every basis has the same number of vectors. Let me write that great fact down.
[35:06]EVERY BASIS, we are given a space, GIVEN A SPACE, R3, or Rn, or some other column space of a matrix, or the nullspace of a matrix or some other vector spaces. Then the great fact is that EVERY BASIS FOR THE SPACE, HAS THE SAME NUMBER OF VECTORS.
[35:53]If one basis has 6 vectors, then every other basis has 6 vectors. So that number 6 is telling me... like telling me how big is the space. It’s telling me how many vectors do I have to have to have a basis. And of course, we’re seeing it this way: that number 6, if we have 7 vectors, we’ve got too many; if we have 5 vectors we haven’t got enough.
[36:22]6 is like just right for whatever space that is. And what do we call that number? That number is... Now I’m ready for the last definition today. It’s the dimension of that space. So every basis for a space has the same number of vectors in it, not the same vectors. All sorts of bases but the same number of vectors, it is always the same. And that number is the DIMENSION. (The video recorder seems skipped several seconds at the point...). This number is the DIMENSION OF THE SPACE. OK.
[37:13]OK, let’s do some examples. But now we got definitions. Let me repeat the four things, the four words that of now we got defined: INDEPENDENCE, that looks at combinations not being 0; SPANNING, that looks at all the combinations, BASIS, that’s the one that combines independence and spanning, and now we’ve got the idea of DIMENSION of a space, it’s a number of vectors in any basis, because all bases have the same number.
[37:40]OK. Let’s take examples. Suppose I take... My space is – examples now – space is the... say the column space of this matrix. Let me write down the matrix. [1, 1, 1; 2, 1, 2; and I’ll just to make it clear, I’ll take the sum, 3, 2, 3; and let me take the sum of all... ah, let me put in 1... yeah, I’ll put in 1, 1, 1] again.
[38:23]OK. So that’s 4 vectors. OK, do they span a column space of that matrix? Let me repeat, do they span the column space of that matrix? Yes! By definition that’s what the column space... where it comes from. Are they the basis for the column space; are they independent?
[38:49]No, they’re not independent. There are something in that nullspace. Maybe we can, so let’s look at the nullspace of the matrix. Tell me a vector that’s in the nullspace of that matrix. So I’m looking for some vector that combines those columns and produces the 0 column.
[39:14]Or in other word, I’m looking for solutions that Ax = 0. So, tell me a vector in the nullspace. Maybe, well this was, this column was that one plus that one, so maybe if I have one of those and minus one of those that would be a vector in the nullspace?
[39:34]So, you’ve already told me now are those vectors independent, the answer is those column vectors... the answer is... no, right? They’re not independent, because... Ah, you knew they weren’t independent. Anyway, -1 of this, -1 of this, +1 of this, 0 of that, is the 0-vector. OK
[39:59]OK. So they are not independent. They span but they are not independent, tell me... a basis, for that column space. What’s the basis for that column space? Here’s the all the questions that the homework asks, the quizzes ask, the final exam will ask: find the basis for the column space of this matrix.
[40:22]OK. Now there’s many answers. But give me the most natural answer. Columns 1 and 2 – that’s the natural answer. Those are the pivot columns, because... I mean we begin systematically, we looked at the first column, it’s OK, we can put that into basis, we look at the second column, it’s OK, we can put that into the basis; the third column we can’t put into basis; the fourth column we can’t again.
[41:00]So the rank of the matrix is? What’s our rank of the matrix? Two! And now that rank is also, we now have another word, we have a great theorem here: THE RANK OF A, that rank r, IS THE NUMBER OF PIVOT COLUMNS, AND IT’S ALSO, well, now please use my new word... This is the number 2, of course, 2. is the rank of my matrix, is the number of pivot columns those pivot columns form a basis of course, so what’s 2?
[41:55]It’s the dimension. The rank of A, the number of pivot columns is the DIMENSION OF THE COLUMN SPACE. Of course you say, it had to be – right, but just look for one moment at the language, the way that English words get involved here. I take the rank of a matrix, the rank of a matrix, it’s the number of (pivot) columns and it’s the dimension of... Not a dimension of a matrix, that’s what I want to say.
[42:43]It’s a dimension of a space, a subspace, the column space. Do you see I don’t take the dimension of A? That’s not what I want. I’m looking for the dimension of the column space of A. If you use those words right, it shows you’ve got the idea right.
[43:04]Similarly here, I don’t talk about the rank of a subspace. It’s a matrix that has a rank. I talk about the rank of a matrix, and the beauty is for these definitions just merge, so that the rank of a matrix is the dimension of its column space, and in this example it’s 2.
[43:24]And then, the further question is what’s the basis? And the first 2 columns are a basis. Tell me another basis, another basis for the column space. You see I just keep hammering away I apologize... I have to be sure you have the idea of basis. Tell me another basis for the column space. Well, you could take columns 1 and 3. That would be a basis for the column space.
[43:55]Or column 2 and 3 would be a basis, or column 2 and 4. Or, tell me another basis that not made out of those columns at all. So, well, I guess I’ve given you infinitely many possibilities. So, I can’t expect a unanimous answer here.
[44:18]I’ll tell you ah... Let’s look at another basis though. I’ll just... Because it’s only 1 out of zillions, I’m gonna put it down, and I’m gonna erase it. Another basis for the column space would be... Ah, let’s see... I’ll put in something that is not there, say... Oh, well, just to make it my life easy (2, 2, 2). That’s in the column space.
[44:43]And that’s sort of obvious. Let me take the sum of those, say (6, 4, 6). Or the sum of all the columns, (7, 5, 7), why not, that’s in the column space. Those are independent, and I’ve got the number right, I’ve got 2. Actually this is a key point. If you know the dimension of the space you’re working with, and we know that this column... we know that the dimension, DIM, the dimension of the column space is 2.
[45:29]If you know the dimension, then... And we have a couple of vectors that are independent; they’ll automatically be a basis. If we got that number of vectors right, 2 vectors in this case, then if they’re independent, they can’t help to span the space.
[45:53]For they didn’t span the space, there be a third guy to help span the space. But they couldn’t be independent. So, it’s just has to be independent, if we got the numbers right... and they span.
[46:08]OK. Very good, so you got the dimension of the space. So this was another basis that I just invented. OK, now I get to ask you about the nullspace. What’s the dimension of the nullspace? So, we got a great fact here. The dimension of the column space is the rank. Now I want to ask you about the nullspace. That’s the other part of the lecture, and it will go on to the next lecture.
[46:44]OK. So, we know the dimension of the column space is 2, the rank. What about the nullspace? This is a vector in the nullspace; are there other vectors in the nullspace? Yes or no.
[47:00]Yes! So this isn’t the basis because it doesn’t span, right? There’s more in the nullspace and we’ve got so far. I need another vector at least. So tell me another vector in the nullspace. Well, the natural choice, you naturally think of is, I’m going on to the fourth column. I’m letting that free variable be a 1 and that free variable be a 0.
[47:29]And I’m asking is that fourth column a combination of my pivot columns? Yes it is. And it’s... that will do! So what I’ve written there? Actually the 2 special solutions, right? I took the 2 free variables, free and free; I gave them the values 1, 0; or 0, 1. I figured out the rest, so... Do you see let me just saying in words: these vectors in the nullspace are telling me... They are telling me the combinations of the columns that give 0.
[48:08]They are telling me in what way the columns are dependent. That’s what the nullspace is doing. Have I got enough now? And what’s the nullspace? Now we have to think about the nullspace. These are 2 vectors in the nullspace they’re independent; are they a basis for the nullspace?
[48:30]What’s the dimension of the nullspace? Do you see that those questions just keep coming up all the time? Are they a basis for the nullspace? You can tell me the answer even though we haven’t written out the proof of that. Can you? Yes or no.
[48:48]Do these 2 special solutions form a basis for a nullspace? In other words, does a nullspace consist of all the combinations of those 2 guys? Yes or no?
[49:00]Yes, yes. The nullspace is 2-dimensional. The nullspace, the dimension of the nullspace is the number of free variables. So the DIMENSION OF THE NULLSPACE IS THE NUMBER OF FREE VARIABLES, and last second, give me a formula. This is then the key formula that we know how many free variables are there? In terms of r the rank, m the number of rows, n the number of columns.
[49:41]What do we get? We have n columns r of them are pivot columns, so, n – r is the number of free columns, free variables and now, it’s the dimension of the nullspace. OK. That’s great. That’s the key spaces, their bases and their dimensions. Thanks!
< Last Modified 1/16/08 5:22 PM |
