About this video
The video recalls that the orthogonal complement of the row space of A is the nullspace of A, and the orthogonal complement of the column space is the left nullspace (the nullspace of A transpose). Using the result that (V perp) perp = V, it shows the reverse: the orthogonal complement of the nullspace is the row space, and the orthogonal complement of the left nullspace is the column space.
You'll be able to
- state that the orthogonal complement of the row space of A is the nullspace of A, and vice versa.
- state that the orthogonal complement of the column space of A is the left nullspace of A, and vice versa.
- use (V perp) perp = V to derive one complement relationship from another.
Key words
- row space
- The span of the rows of A, which is the same as the column space of A transpose.
- nullspace
- The set of all vectors x with Ax = 0.
- left nullspace
- The nullspace of A transpose, which is the orthogonal complement of the column space of A.
- orthogonal complement
- All vectors orthogonal to every vector in a given subspace.
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.
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