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

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!

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!

@fresh_42 Right! Thank you for bringing that to my attention!