What Geometrical Objects Do Subspaces of V3(R) Represent?

[ashnicholls]

Here is a question I have been given:

V3(R) represents the set of vectors in 3-dimensional space. What kind of geometrical objects are represented by the various subspaces of V3(R)? For instance the 1-dimensional subspace S with basis { (0, 1, 0)T } represents the set of vectors parallel to the y-axis, so the set of points with position vectors in S is the y-axis itself. Since any 3-dimensional subspace of V3(R) is V3(R) itself, you need only consider subspaces of dimension less than 3. You should find that the range of different kinds of geometrical object represented by the subspaces of V3(R) is quite restricted.

I do not know what this is asking.

Does it mean looking at planes?

But surely there is more to the question than that?

Has anyone got any clues or tips.

Cheers

 

[HallsofIvy]

Here is a question I have been given:

V3(R) represents the set of vectors in 3-dimensional space. What kind of geometrical objects are represented by the various subspaces of V3(R)? For instance the 1-dimensional subspace S with basis { (0, 1, 0)T } represents the set of vectors parallel to the y-axis, so the set of points with position vectors in S is the y-axis itself. Since any 3-dimensional subspace of V3(R) is V3(R) itself, you need only consider subspaces of dimension less than 3. You should find that the range of different kinds of geometrical object represented by the subspaces of V3(R) is quite restricted.

I do not know what this is asking.

Does it mean looking at planes?

But surely there is more to the question than that?

Has anyone got any clues or tips.

Cheers

Any one-dimensional subspace of R³ is a straight line through the origin. Any two-dimensional subspace of R³ is a plane containing the orgin. Of course, the only three-dimensional subspace of R³ is R³ itself.

You can think of the set containing only the 0 vector itself as being a zero-dimensional subspace- although some texts refuse to allow that as a vector space.

 

[ashnicholls]

Yes ok thank, that is what I thought it roughly was, but what is the question asking?

Cheers

 

[matt grime]

For you to describe the possible subspaces. They are planes, or lines, and must pass through the origin. What else could it be asking you to write down?

 

[ashnicholls]

O that's just seems very simple?

Cheers

 

[matt grime]

There's no reason why every question has to be fiendishly hard. Just ask yourself if you've answered the question to the best of your ability - that is all you can ever do.