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.