Hyperplanes H1 and H2 have dimensions p and q

[JG89]

This is a question in my textbook:

"The hyperplanes H1 and H2 have dimensions p and q, respectively. What is the smallest dimension which the hyperplane H3 must have in order to be sure to contain both H1 and H2?"

I reasoned it out like this.

A basis for H1 would be x1 + x2 +...+ xp

And a basis for H2 would be xp+1 + xp+2 +...+xq

So, a hyperplane which would contain all of these vectors would have the basis:

x1 + x2 +...+ xp + xp+1 +...+ xq

So its dimension would be p+q. However, the answer in the back of my book says the answer is p + q + 1. What is wrong with what I did?

 

[fopc]

This is a question in my textbook:

"The hyperplanes H1 and H2 have dimensions p and q, respectively. What is the smallest dimension which the hyperplane H3 must have in order to be sure to contain both H1 and H2?"

I reasoned it out like this.

A basis for H1 would be x1 + x2 +...+ xp

And a basis for H2 would be xp+1 + xp+2 +...+xq

So, a hyperplane which would contain all of these vectors would have the basis:

x1 + x2 +...+ xp + xp+1 +...+ xq

So its dimension would be p+q. However, the answer in the back of my book says the answer is p + q + 1. What is wrong with what I did?

In the given context, I'd say a hyperplane of vector space V, where dimV = n, is an (n-1)-dimensional linear manifold.
From here you should be able to arrive at the answer given in the back of the book.

 

[HallsofIvy]

Saying that a hyperplane in V must have dimension n-1 would make the question non-sense: it would then make no sense to say H1 has dimension "p" and H2 has dimension "q" if they both must have dimension n-1.

I think here "hyperplane" just means a "linear manifold" mentioned by fopc, of any dimension: start with a subspace and add some fixed vector, not in the subspace, to every vector. In 3 dimensions, for example, a plane, NOT containing the origin, is a linear manifold but not a subspace. In that case, we cannot talk about a "basis" for a hyperplane.

We can "move" the hyperplane to the origin: choose any vector in the hyperplane and subtract it from each vector in the hyperplane, reversing the "add some fixed vector" I mentioned above.

For example, in 3 dimensions, we can think of two skew lines as linear manifolds so p= q= 1. (If we were talking about subspaces, we would have to have two lines intersecting at the origin so they couldn't be skew.) That will require 3 dimensions to include both, not just 2.

 

[fopc]

Saying that a hyperplane in V must have dimension n-1 would make the question non-sense: it would then make no sense to say H1 has dimension "p" and H2 has dimension "q" if they both must have dimension n-1.

[snip]

I said nothing of the sort.

Nobody said (or even suggested) "they both must have dimension n-1".

I suggest you try reading the post again--carefully this time.

If he's given that one hyperplane has dimension p, then clearly it resides in a p+1 dimensional vector space. That would clearly follow from what I *did* say in my post.
Same for q, it sits in a q+1 dimensional vector space.
From this he should be able to see how to proceed.

EDIT: Forget it. I can see this is a waste of time. Adios.

 

[HallsofIvy]

I said nothing of the sort.

Nobody said (or even suggested) "they both must have dimension n-1".

I suggest you try reading the post again--carefully this time.

If he's given that one hyperplane has dimension p, then clearly it resides in a p+1 dimensional vector space. That would clearly follow from what I *did* say in my post.
Same for q, it sits in a q+1 dimensional vector space.
From this he should be able to see how to proceed.

EDIT: Forget it. I can see this is a waste of time. Adios.

Then they both must be in some higher dimensional space, which is what I said. I don't see how arguing about some "p+1 dimensional space" helps.

 

[JG89]

Saying that a hyperplane in V must have dimension n-1 would make the question non-sense: it would then make no sense to say H1 has dimension "p" and H2 has dimension "q" if they both must have dimension n-1.

I think here "hyperplane" just means a "linear manifold" mentioned by fopc, of any dimension: start with a subspace and add some fixed vector, not in the subspace, to every vector. In 3 dimensions, for example, a plane, NOT containing the origin, is a linear manifold but not a subspace. In that case, we cannot talk about a "basis" for a hyperplane.

We can "move" the hyperplane to the origin: choose any vector in the hyperplane and subtract it from each vector in the hyperplane, reversing the "add some fixed vector" I mentioned above.

For example, in 3 dimensions, we can think of two skew lines as linear manifolds so p= q= 1. (If we were talking about subspaces, we would have to have two lines intersecting at the origin so they couldn't be skew.) That will require 3 dimensions to include both, not just 2.

Thanks. I get it now.