Proof of Proposition 2 in Ch.7 Spivak Vol.2: n-Dimensional Distribution?

[joe2317]

Hi.

I have a question on proof of proposition 2 in chater 7 Spivak volume2.
In the proof, he says that the n-dimensional distribution
$$\Delta_{p}$$=$$\bigcap^{n}_{i,j=1}$$ker$$\Lambda^{i}_{j}$$(p)
in R^(n+n^2) is integrable.
Could anyone explain why it is an n dimensional distribution?
Thanks.

 

[joe2317]

Never mind. I figured this out.

 

[tacman]

Sure, I'd be happy to provide a response.

First, let's define what is meant by an "n-dimensional distribution". In general, a distribution on a manifold M is a subspace of the tangent space at each point that varies smoothly with the point. In other words, it assigns a subspace of the tangent space at each point in a smooth way.

Now, in the case of the n-dimensional distribution in question, we are considering a distribution on the manifold R^(n+n^2), which is the space of n x n matrices. This space has dimension n + n^2, and we are interested in finding a subspace of dimension n at each point that varies smoothly with the point.

The key to understanding why the intersection of the kernels of the n x n matrices is an n-dimensional distribution lies in the definition of the kernel of a matrix. The kernel of a matrix is the set of all vectors that are mapped to the zero vector under the action of the matrix. In other words, it is the set of vectors that the matrix "flattens" to zero.

Now, in the proof of Proposition 2, Spivak shows that the intersection of the kernels of the matrices \Lambda^{i}_{j}(p) is precisely the set of all tangent vectors at p that are "flattened" to zero by all of the matrices. In other words, it is the set of all tangent vectors at p that are orthogonal to all of the rows and columns of all of the matrices. This subspace has dimension n, as it is the intersection of n subspaces of dimension n.

Furthermore, Spivak shows that this subspace varies smoothly with the point p, which is the key requirement for a distribution. Therefore, we can conclude that the intersection of the kernels of the matrices \Lambda^{i}_{j}(p) is indeed an n-dimensional distribution on R^(n+n^2).

I hope this helps to clarify why this is the case. Please let me know if you have any further questions.