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# Orthogonal complement of the nullspace

Video transcript

I've got some matrix A. We learned several videos ago
that it's row space is the same thing as the column space
of it's transpose. So that right there is
the row space of A. That this thing's orthogonal
complement, so the set of all of the vectors that are
orthogonal to this, so its orthogonal complement is equal
to the nullspace of A. And, essentially, the same
result if you switch A and A transpose, we also learned that
the orthogonal complement of the column space of A
is equal to the left nullspace of A. Which is the same thing as the
nullspace of A transposed. We could write this, just to
understand the terminology, that's the left nullspace, which
is the same thing as the nullspace of A transposed. Now, what is the orthogonal
complement of the nullspace of A? Well, you might guess that it's
the row space of A, but we didn't have the tools
until the last video to figure that out. In the last video, we saw that
if we take the orthogonal complement -- let me write it
this way -- if we were to take the orthogonal complement of the
orthogonal complement, it equals the original sub space. So now, what are we doing? We're taking the orthogonal
complement of the nullspace of A. Well, the nullspace of A is just
this thing right here. So this is equal to taking the
orthogonal complement of the nullspace of A. But the nullspace of
A is this thing. It's the row space's orthogonal
complement. Now, we're essentially the
orthogonal complement of the orthogonal complement. We can use this property, which
we just proved in the last video, to say that this
is equal to just the row space of A. Which is the same thing as the
column space of A transposed. So the orthogonal complement
of the row space is the nullspace and the orthogonal
complement of the nullspace is the row space. We can apply that
same property on this side right here. What is the orthogonal
complement of the left nullspace of A? What is this? Well, this is going to be
equal to the orthogonal complement of this thing. Because that's what the left
nullspace of A is equal to. So it's equal to the orthogonal
complement of the orthogonal complement
of the column space. And we just learned in the last
video, if you take the orthogonal complement of the
orthogonal complement it equals the original subspace. So this is just equal to
the column space of A. So we now see some
nice symmetry. The nullspace is the orthogonal
complement of the row space, and then we see
that the row space is the orthogonal complement
of the nullspace. Similarly, the left nullspace
is the orthogonal complement of the column space. And the column space is the
orthogonal complement of the left nullspace. So we have some nice symmetry
that we're able to essentially prove given what we saw
in the last video.