Smallest subspace if a plane and a line are passing through the origin

In summary, the smallest vector space containing both P and L is the span of P and L, unless L lies on P in which case the smallest subspace is L.
  • #1
sindhuja
3
2
Hi all, I am a beginner in Linear Algebra. I am solving problems on vector spaces and subspaces from the book Introduction to Linear Algebra by Gilbert Strang. I have come across the following question:

Suppose P is a plane through (0,0,0) and L is a line through (0,0,0). The smallest vector space containing both P and L is either ______ or ______.

My understanding: If L lies on P, then the smallest subspace is L. If L does not lie on P, then the smallest subspace is the zero vector (0,0,0). I am aware I am missing something here. Could someone please clarify what I am missing? Thanks in advance!
 
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  • #2
sindhuja said:
Hi all, I am a beginner in Linear Algebra. I am solving problems on vector spaces and subspaces from the book Introduction to Linear Algebra by Gilbert Strang. I have come across the following question:

Suppose P is a plane through (0,0,0) and L is a line through (0,0,0). The smallest vector space containing both P and L is either ______ or ______.

My understanding: If L lies on P, then the smallest subspace is L.
Think again. We want P AND L to be included.
sindhuja said:
If L does not lie on P, then the smallest subspace is the zero vector (0,0,0).
Same problem. We are talking about a space that is spanned by P and L. You are talking about their intersection.
sindhuja said:
I am aware I am missing something here. Could someone please clarify what I am missing? Thanks in advance!
 
  • #3
@fresh_42 Right! Thank you for bringing that to my attention!
 

1. What is the definition of a subspace?

A subspace is a subset of a vector space that is closed under addition and scalar multiplication. This means that any linear combination of vectors in the subspace will also be in the subspace.

2. How do you determine the smallest subspace if a plane and a line are passing through the origin?

The smallest subspace that contains a plane and a line passing through the origin is the intersection of the two subspaces. This is because the intersection will contain all of the vectors that are in both the plane and the line, making it the smallest subspace that satisfies both conditions.

3. Can a subspace contain only one vector?

Yes, a subspace can contain only one vector. This vector would be considered a basis for the subspace, as it is the only vector needed to span the subspace.

4. How many dimensions does the smallest subspace containing a plane and a line have?

The smallest subspace containing a plane and a line passing through the origin will have at least two dimensions, as it must contain both the plane and the line. However, it is possible for the intersection of the two subspaces to have more than two dimensions if the plane and line are not parallel.

5. Is the smallest subspace containing a plane and a line always unique?

No, the smallest subspace containing a plane and a line passing through the origin is not always unique. It depends on the orientation and position of the plane and line. If the plane and line are parallel, then the intersection will be the line itself, resulting in a one-dimensional subspace. However, if the plane and line are not parallel, then the intersection will be a two-dimensional subspace.

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