MIT18_06S10_L11

MIT18_06S10_L11 - 18.06 Linear Algebra Spring 2010...

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18.06 Linear Algebra, Spring 2010 Transcript – Lecture 11 OK. This is linear algebra lecture eleven. And at the end of lecture ten, I was talking about some vector spaces, but they're -- the things in those vector spaces were not what we usually call vectors. Nevertheless, you could add them and you could multiply by numbers, so we can call them vectors. I think the example I was working with they were matrices. So the -- so we had like a matrix space, the space of all three by three matrices. And I'd like to just pick up on that, because -- we've been so specific about n dimensional space here, and you really want to see that the same ideas work as long as you can add and multiply by scalars. So these new, new vector spaces, the example I took was the space M of all three by three matrices. OK. I can add them, I can multiply by scalars. I can multiply two of them together, but I don't do that. That's not part of the vector space picture. The vector space part is just adding the matrices and multiplying by numbers. And that's fine, we stay within this space of three by three matrices. And I had some subspaces that were interesting, like the symmetric, the subspace of symmetric matrices, symmetric three by threes. Or the subspace of upper triangular three by threes. Now I, I use the word subspace because it follows the rule. If I add two symmetric matrices, I'm still symmetric. If I multiply two symmetric matrices, is the product automatically symmetric? No. But I'm not multiplying matrices. I'm just adding. So I'm fine. This is a subspace. Similarly, if I add two upper triangular matrices, I'm still upper triangular. And, that's a subspace. Now I just want to take these as example and ask, well, what's a basis for that subspace? What's the dimension of that subspace? And what's bd- dimension of the whole space? So, there's a natural basis for all three by three matrices, and why don't we just write it down. So, so M, a basis for M. Again, all three by threes. OK. And then I'll just count how many members are in that basis and I'll know the dimension. And OK, it's going to take me a little time. In fact, what is the dimension? Any idea of what I'm coming up with next? How many numbers does it take to specify that three by three matrix? Nine. Nine is the, is the dimension I'm going to find. And the most obvious basis would be the matrix that's that matrix and then this matrix with a one there and that's two of them, shall I put in the third one, and then onwards, and the last one maybe would end with the one. OK. That's like the standard basis.
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In fact, our space is practically the same as nine dimensional space. It's just the nine numbers are written in a square instead of in a column. But somehow it's different and, and ought to be thought of as -- natural for itself.
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