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Okay. This is the Neralgebra, lecture 11. And at the end of lecture 10, I was talking
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about some vector spaces, but the things in those vector spaces were not what we usually
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call vectors. Nevertheless, you could add them and you could multiply by numbers. So we
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can call them vectors. I think the example I was working with, they were matrices. So
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we had like a matrix space, the space of all three by three matrices. And I'd like to
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just pick up on that because we've been so specific about n-dimensional space here. And
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you really want to see that the same ideas work as long as you can add and multiply by
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the matrices. So these new, new vector spaces, the example I talked was the space m of all
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three by three matrices. Okay. I can add them, I can multiply by scalars. I can multiply
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two of them together, but I don't do that. That's not part of the vector space picture.
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The vector space part is just adding the matrices and multiplying by numbers. And that's fine.
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We stay within this space of three by three matrices. And I had some subspaces that were
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interesting like the symmetric, the subspace of symmetric matrices, symmetric three by
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three, or the subspace of upper triangular three by three. Now I use the word subspace
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because it follows the rule. If I add two symmetric matrices, I'm still symmetric. If
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I multiply through symmetric matrices, is the product automatically symmetric? No. But
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I'm not multiplying matrices. I'm just adding. So I'm fine. This is a subspace. Similarly,
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if I add two upper triangular matrices, I'm still upper triangular. And that's a subspace.
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Now I just want to take these as examples and ask, well, what's the basis for that subspace?
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What's the dimension of that subspace? And what's the dimension of the whole space? So there's
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a natural basis for all three by three matrices. And why don't we just write it down? So,
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m, a basis for m, again, all three by three. And then I'll just count how many members are
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in that basis, and I'll know the dimension. And okay, it's going to take me a little time.
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In fact, what is the dimension? Any idea of what I'm coming up with next? How many
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numbers does it take to specify that three by three matrix? 9. 9 is the dimension I'm going
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to find. And the most obvious basis would be the matrix sets, that matrix. And then this
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matrix with a one there. And that's two of them. Shall I put in the third one? And then onwards.
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And the last one maybe would end with the one. Okay. That's like the standard basis. In fact,
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our space is practically the same as nine dimensional space. It's just the nine numbers are written
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in a square instead of in a column. But somehow it's different and ought to be thought of as
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sort of natural for itself. Because now what about the symmetric three by three? So, that's a
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subspace. Just let's just think what's the dimension of that subspace and what's a basis for that
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subspace? Okay. And I guess this question occurs to me. If I look at this subspace of symmetric
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three by three's. Well, how many of these original basis members belong to the subspace? I think
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only three of them do. This one is symmetric. This last one is symmetric. And the one in the
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middle with a one in that position in the two two position would be symmetric. But that so I've
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got three of these original nine are symmetric. But so this is an example where but that's not
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all right. And what's the dimension? Let's put the dimension down. Dimension of M was not.
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What's the dimension of so we call this S is what? What's the dimension of this? I'm sort of
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taking simple examples where we can we can spot the answer to these questions. So how many
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if I have a symmetric think of all symmetric matrices as a subspace? How many parameters do I
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choose in three by three symmetric matrices? Six. Right. If I choose the diagonal that's three
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and the three entries above the diagonal then I know what the three entries below. So the dimension
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is six. I guess what's the dimension of this here? Let's call this space U from upper triangular.
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So what's the dimension of that space of all upper triangular three by three? Again six.
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And but we haven't got we haven't seen a bit. Well actually maybe we have got a basis here for
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the upper triangular. I guess six of these guys. One two three four and a couple more would be
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upper triangular. So there's an accidental case where the big basis contains in it a basis for
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the subspace. But what the symmetric guy didn't have. The symmetric guy the basis so you see
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what a basis is the basis for the big space we generally need to think it all over again to get a
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basis for the subspace. And then how do I get other subspaces? Well we spoke before about the
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subspace the symmetric matrices and the upper triangle. This is symmetric and upper triangle.
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Okay. What's the dimension of that space? What's in that space? So what's the matrix
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is symmetric and also upper triangular? That makes it diagonal. So this is the same as the diagonal
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matrices diagonal three by three. And the dimension of this of s intersect U. Right? You're okay
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with that symbol? That's the vectors that are in both s and U and that's D. So s intersect U is
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the diagonals and the dimension of the diagonal matrices is three. And we've got a basis no problem.
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Okay. As I write that I think okay what about putting to get so this is like this intersection is
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taking all the vectors that are in both that are that are symmetric and also upper triangle.
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Now we looked at the union. Suppose I take the matrices that are symmetric or upper triangle.
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Why was that no good? Why am I not interested in the union putting together those two subspaces?
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So these are matrices that are in s or in U or possibly both. So the diagonals included. But what's
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bad about this? It's not a subspace. It's like having taking a couple of lines in the plane and
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stopping there. Align this is so there's a three-dimensional subspace of a nine-dimensional space.
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There's six. There's a six-dimensional subspace of a nine-dimensional space. There's another one.
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But they they're headed in different directions. So we can't just put them together. We have to fill
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in. So that's what we do. To get this bigger space that I'll write with a plus sign this is combinations
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of things in s and things in U. Okay. So that's that's the final space I'm going to introduce.
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I have a couple of subspaces. I can take their intersection and now I'm interested in not their union
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but their sum. So this would be the this is the intersection and this will be their sum.
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So what what do I need for a subspace here? I take anything in s plus anything in U.
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I don't just take things that are in s and pop in also separately things that are in U.
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This is the sum of any element of s that is any symmetric matrix plus any
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in U any element of U. Okay. Now as long as we got an example here tell me what we get.
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Okay. If I take every symmetric matrix take all symmetric matrices and add them to all
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upper triangular matrices then I've got a whole lot of matrices and it is a subspace.
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And what's that what or it's a vector space and what vector space would I then have?
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Any idea what what matrices can I get out of a symmetric plus an upper triangle?
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I can get anything. I get all matrices. I get all three by three.
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What's worth thinking about that? It's just like stretch your mind a little just a little
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to to think of these subspaces and what their intersection is and what their sum is.
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And now can I give you a little oh let's figure out the dimension. So what's the dimension
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of s plus U in this example is?
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No because we got all three by three. So the original spaces had the original symmetric space
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had dimension six and the original upper triangular space had dimension six. And actually
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I'm seeing here a nice formula that the dimension of s plus the dimension of U
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if I have two subspaces the dimension of one plus the dimension of the other equals.
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The dimension of their intersection plus the dimension of their sum. Six plus six is three plus not.
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It's kind of satisfying that these natural operations and we these this is the effect.
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This is the set of natural things to do with with sub-paces that the dimensions come out
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in a good way. Okay maybe I'll take just one more example of a vector space that doesn't have
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vectors in it. It's say take come from differential equations. So this is a one more new vector space
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that will give just a few minutes to. Suppose I have a differential equation like V second Y,
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the X squared plus Y equals zero. Okay I look at the solutions to that equation.
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So what are the solutions to that equation? Y equals cos X is a solution. Y equals sine X is a
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solution. Y equals well e to the I X is a solution if you want if you allow me to put that in.
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Oh but why should I put that in? It's already there. You see I'm really looking at a null space
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here. I'm looking at the null space of a differential equation. That's the solution space.
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And describe the solution space, all solutions to this differential equation. So the equation is Y double
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prime plus Y equals zero. Cosines the solution. Sin is a solution. Now tell me all the solutions.
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So I don't need e to the I X forget that. What are all the complete solutions?
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Is what? A combination of these. The complete solution is Y equals some
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multiple of the cosine plus some multiple of the sine. That's a vector space.
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That's a vector space. What's the dimension of that space? What's a basis for that space?
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Okay let me ask you a basis first. If I take the set of solutions to that second order differential
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equation, there it is. Those are the solutions. What's a basis for that space?
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Now remember what's the what question am I asking? Because if you know the question I'm asking
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you'll see the answer. A basis means all the guys in the space are combinations of these basis
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vectors. Well this is a basis. Sin X plus X there is a basis. Those two, they're like the special
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solution. We had special solutions the A X equal B. Now we've got special solutions to differential
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equations. Sorry we had special solutions to A X equals zero I misspoke. Special solutions
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were for the null space. Just as here we're talking about the null space. Do you see that here is
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those two and what's the dimension of the solution space is how many vectors in this basis?
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Two, the sine and cosine. Are those the only basis for this space? By no means. E to the ix
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and E to the minus ix would be another basis. Lots of basis. But you see that really what a
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course in differential in linear differential equations is about is finding a basis for the
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solution space. The dimension of the solution space will always be two because we have a second
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order equation. So that's like there's 1803 and five minutes of 1806 is enough to take care of
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1803. Okay so there's a that's one more example and of course the point of the example is these things
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don't look like vectors. They look like functions but we can call them vectors because we can add them
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and we can multiply by constants so we can take linear combinations. That's all we have to be
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allowed to do. So that's really why this idea of linear algebra and basis and dimension and so on
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plays a wider role than then our constant discussions of m by m matrices. Okay that's what I wanted
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to say about that topic. Now of course the key number associated with matrices to go back to
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that number is the rank. And the rank what do we know about the rank? Well we know it's not
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bigger than m and it's not bigger than m. So but I'd like to have a little discussion on the rank.
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Maybe I'll put that here. So I'm picking up this topic of rank one matrices. And the reason I
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interested in rank one matrices is nothing ought to be simple. If the rank is only one the matrix
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can't get away from us. So for example let me take let me create a rank one matrix. Okay suppose
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it's three but suppose it's two by three. And let me give you the first row.
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What's what can the second row be? Tell me a possible second row here for this matrix to have
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rank one. A possible second row is two eight ten. The second row is a multiple of the first row.
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It's not independent. So tell me a basis for the oh yeah sorry to keep bringing up these same
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questions after the quiz I'll stop but for now tell me a basis for the row space. A basis for
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the row space of that matrix is the first row right the first row one four five. A basis for the
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column space of this matrix is what's the dimension of the column space? The dimension of the column
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space is also one right because it's also the rank. The dimension you remember the dimension
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of the column space equals the rank equals the dimension of the column space of the transpose
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which is the row space of A. Okay and in this case it's one R and one.
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And sure enough all the columns are all the other columns are multiples of that column. Now
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there's there ought to be a nice way to see that and here it is. I can write that matrix as
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it's pivot column one two times it's one four five. A column times a row one column times one
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row gives me a matrix right if I multiply a column by a row that's two by one matrix times a one
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by three matrix and the result of the multiplication is two by three and it comes out right. So
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what I want to make my point is the rank one matrices that every rank one matrix has the form
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sum, sum, sum, sum, row. So U is a column vector, V is a column vector but I make it into a row by
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putting in V transpose. So that's the that's the that's the complete picture of rank one
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matrices. We'll be interested in rank one matrices. Later we'll find oh they're determinant that
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will be easy, they're eigenvalues that'll be interesting. Uh, rank one matrices are like the
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building blocks for all matrices and actually maybe you can guess. And if I took any matrix
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a five by 17 matrix of rank four then it seems pretty likely and it's true that I could break
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that five by 17 matrix down as a combination of rank one matrices and probably how many of
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those would I need. If I have a five by 17 matrix of rank four I'll need four.
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Four rank one matrices. So the rank one matrices are the building blocks and how I can produce
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every I can produce every five by every rank four matrix out of four rank one matrices.
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Okay, that brings me to a question of course. Would the rank four matrices form a subspace?
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Let me take all five by 17 matrices and think about rank four matrices. The subset of rank four
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matrices. Let me all write this down. You see I'm I'm reviewing for the quiz section because I'm
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asking the kind of questions that are short enough but to bring out do you know what these words
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are? So I take my matrix space and now it's all five by 17 matrix. And now the
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subset the question I asked is subset of rank four matrices. Is that a subspace? If I add a matrix
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up so if I multiply a matrix of rank four by a rank four or less let's say because I have to let
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the zero matrix in. If it's going to be a subspace but that doesn't just because the zero matrix
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got in there doesn't mean I have a subspace. So if I so the question really comes down to if I add
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two rank four matrices is the sum rank four? What do you think? No, not usually. Not usually.
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If I add two rank four matrices the sum is probably what could I say about the sum? Well actually
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well the rank could be five. It's a general fact actually that the rank of a plus b can't be more
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than rank of a plus the rank of b. So this would say if I add two of those the rank couldn't be
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lower than eight but I know actually the rank couldn't be as large as eight anyway. How big for the
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rank b for the rank of a matrix in him could be as large as five. So that they're all sort of
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natural ideas. So rank four matrices or rank one matrices. Let me say that's a rank one.
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Let me take the subset of rank one matrices. Is that a vector space?
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If I add a rank one matrix or a rank one matrix no it's most likely going to have rank two. So this
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is so much that's make that point not a subset. Okay. Okay, those are topics that I wanted to
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just fill out the previous lectures. I'll ask one more subspace question a more a more
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likely example. Suppose I'm in let me put this example on a new board.
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Suppose I'm in our in our form. So my typical my my typical vector in our form at four
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components the one, the two, the three and the four. Suppose I say the subspace of vectors
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whose components add to zero. So I let s be all b all vector b in four dimensional space
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with the one plus b two plus b three plus b four equals zero.
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So I just want to consider that bunch of vectors if it is subspace first of all.
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It is a subspace. It is a subspace. What's how do we see that?
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It is a subspace. I formally I could check if I add one vector that
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with whose component add to zero and I multiply that vector by six the components still add to zero
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just six times six times zero. If I have a couple of v and w and I add them the components still
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add to zero. Okay, it's a subspace. What's the dimension of that space and what's the basis for that space?
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So you see how I can just describe a space and we can ask for the dimension at the basis first
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and the dimension. Of course the dimension is the one that's easy to tell me in a single word.
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What's the dimension of our subspace s here?
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And the basis. Tell me some vectors in it.
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Well I'm going to ask you again to guess the dimension. Again I think I heard it.
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Three. The dimension is three. How does this connect to our a x equal zero? Is this the
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null space of something? Is that the null space of the matrix and then we can look at the matrix
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and we know everything about those sub spaces. This is the null space of what matrix?
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What's the matrix where the null space is then a b equal zero. So I'm on this to phrase you to be
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a b equal zero. B is now the vector and what's the matrix that we're seeing there?
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It's the matrix of four one. You see that that's if I don't get a b equal zero for this matrix
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a I multiply by b and I get this requirement because the component said zero. So I'm really
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going to speak about s. I'm speaking about the null space of that matrix.
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Okay let's just say we've got a matrix now. We want it to null space.
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Well tell me it's ranked first. The rank of that matrix is
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one. Thanks. The r is one. What's the general formula for the dimension of the null space?
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The dimension of the null space of a matrix is in general an m by n matrix of rank r.
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How many independent guys in the null space? And minus all right? And minus r. In this case n is four.
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Four problems. The rank is one so the null space is free to make. So of course you can see it in this
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case but you can also see it here in our systematic way of dealing with the four fundamental sub
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spaces of a matrix. So what actually what are all four sub spaces? The row space is clear.
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The row space is in r four. Can we take the four fundamental sub spaces of this matrix?
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This is kill this example. The row space is one dimensional. It's all multiple of that of that row.
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The null space is free to make. Oh you better give me a basis for the null space.
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So what's a basis for the null space? The special solution. To find the special solutions I look for the three
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variables. The three variables here are their specific. The three variables are on the two
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principles. So the basis for F or S will be I'm expecting three vectors. Three special solutions. I give the value
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one to that three variable and what's the pivot variable if this is going to be a vector in S minus one.
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Now the row is at zero. The entry is at zero. The second special solution has a one the second
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free variable and again a minus one makes it run. The third one and the one and the third free
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variable and again a minus one makes it run. That's my answer. That's the answer I would be looking for.
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The basis for the sub space S you would just list three vectors and those would be the natural
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three to list. Not the only possible three but those are the special three. Okay tell me about
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the column space. What's the column space of this matrix? A.
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So the column space is a sub space of R one because N is only one. The column is only
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have one component. So the column space of S, the column space of A is somewhere in the space R one because we only have these columns are short.
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And what is the column space actually? I just it's just talking with these words is what I'm doing.
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The column space for that matrix is R one. The column space for that matrix is almost a pose of that.
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Oh there it is. And all of us give you all of our one.
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And what's the remaining fourth space in all space of any transform? In what?
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So we transpose A. We look for combinations of the columns now that give zero for A transpose
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and there aren't any. The only thing the only combination of these rows to give the zero row is zero.
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Okay so let's just check dimensions. The null space has dimension three. The row space has
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dimension one three plus one is four. The column space has dimension one and what's the dimension of this
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like smallest possible space? What's the dimension of the zero space? It's a sub space.
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Zero. What else could it be? I mean let's we have to take a reasonable answer and the only reasonable answer is zero.
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So one plus zero gives this was n. The number of columns and this is n the number of rows.
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And let's just let me just say again then the the the sub space that has only that one point that
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point is zero dimension of course. And the basis is empty because if the dimension is zero there
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shouldn't be anybody in the basis. So the basis of that smallest sub space is the empty set
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and the number of members in the empty set is zero so that's the dimension. Okay.
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Good. Now I have just five minutes to tell you about well actually about some some some
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this is now this last topic of small world graph and leads into a lecture about graphs and maybe
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a relatively rough. But let me tell you the in these last minutes the graph that I'm interested in.
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It's the graph where so what is a graph? Better tell you that first. Okay. What's a graph?
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Okay. This isn't calculus. We're not I'm not thinking of like some sin curve.
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The word graph is used in a completely different way. It's a set of a bunch of nodes and edges.
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Ages connecting the notes. So I have nodes like five nodes and edges I'll put in some edges.
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I could put in through them all. There's well I'm gonna put in a couple more.
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There's a graph with five nodes and one two three four five six edges.
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And some five by six matrix is going to tell us everything about that graph.
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Let me leave that matrix the next time and tell you about the question I'm interested in.
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Suppose the graph isn't just doesn't have this five nodes but suppose every
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suppose every person in this room is a node.
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And suppose there's an edge between two nodes if those two people are friends.
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So have I described the graph? It's a pretty big graph hundred nodes and I don't know how many edges they're in there.
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There's an edge if your friend. So that's the graph for this class.
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A similar graph you could take for the whole country. So 260 million nodes and edges between friends.
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And the question for that graph is how many steps does it take to get from anybody to anybody?
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What two people are furthest apart in this friendship graph? Say for the US.
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By furthest apart I mean the distance from well I'll tell you my distance to Clinton.
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It's true. I happen to go to college with somebody who knows Clinton.
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I don't know. So my distance to Clinton is not one because I don't happily or not.
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I don't know him. But I know somebody who does. He's a senator and so I presume he knows.
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Okay I don't know what your what's your distance to Clinton.
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Well not more than three right actually true.
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You know me I take credit for reducing your distance to what's your distance to Monica.
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Not anybody below below four is in trouble here.
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Or maybe three but right.
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So what's Hillary's distance to Monica? I don't think you'd better put that on tape here.
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One or two I think.
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Is that right?
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Well we won't say more about that.
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Actually the real question is what's one of big large distances?
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How far apart could people be separated?
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And roughly this number six degrees of separation is kind of appeared.
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It's a movie title. It's a book title.
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And it's with this meaning that roughly speaking six might be a fairly...
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Not too many people.
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If you sit next to somebody on airplane you get talking to them.
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You begin to discuss mutual friends to sort of find out okay what connections do you have.
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And very often you'll find you're connected in my two or three or four sets.
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And you remark it's a small world.
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And that's how this expression is small world in the thing.
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But six, I don't know if you've been fine.
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If it took six I don't know if you would successfully discover those six in an airplane
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conversation.
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But here's the math question.
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And I'll leave it for next for lecture 12 and do a lot of linear algebra in lecture 12.
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But the interesting point is that with a few shortcuts the distances come down dramatically.
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That all your distances to Clinton immediately dropped to three by taking linear algebra.
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That's like a vector of both or taking linear algebra.
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And to understand mathematically what it is about these graphs or like the graphs of the
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worldwide web.
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There's a fantastic graph.
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So many people would like to understand and model the web.
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What are the... where the edges are links and the nodes are our sites, websites.
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I'll leave you with that graph.
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And see you have a good weekend and see you on Monday.