What are the types of subspaces in R^4?

[rocomath]

There are more answers to this problem, but I'm not sure how to approach it.

The subspaces of $$R^3$$ are planes, lines, $$R^3$$ itself, or Z containing only (0,0,0,0).

b. Describe the five type of subspaces of $$R^4$$

i. lines thru (0,0,0,0)
ii. zero (0,0,0,0)

iii. planes thru (0,0,0,0)
What do you mean by planes? Are you assuming 2 dimensional sets as is usual with planes?

iv. itself, $$R^4$$

What's the 5th?

 

[HallsofIvy]

There are more answers to this problem, but I'm not sure how to approach it.

The subspaces of $$R^3$$ are planes[/quote]
Planes containing the origin

, lines

Lines through the origin

, $$R^3$$ itself, or Z containing only (0,0,0,0).

caution: Z is typically used to mean the set of integers. Here, I guess you mean "the set containing only (0, 0, 0)". (NOT (0,0,0,0)!)

b. Describe the five type of subspaces of $$R^4$$

i. lines thru (0,0,0,0)
ii. zero (0,0,0,0)

More correctly, the set containing only the zero vector.

iii. planes thru (0,0,0,0)

What do you mean by planes? Are you assuming 2 dimensional sets as is usual with planes?

iv. itself, $$R^4$$

What's the 5th?

Look at number iii again. You have incorporated two different "kinds" of subspace there.
(R⁴ is 4 dimensional. Think of your five kinds of subspace as: 0, 1, 2, 3, 4.)

 

[rocomath]

Look at number iii again. You have incorporated two different "kinds" of subspace there.
(R⁴ is 4 dimensional. Think of your five kinds of subspace as: 0, 1, 2, 3, 4.)

iii. planes thru (0,0,0,0)

iv. itself, Click to see the LaTeX code for this image

What's the 5th?

So I have planes in R^4. But I also have planes in R^2 and R^3 passing thru (0,0,0,0) that are contained in R^4. Lastly, R^4, itself.

 

[matt grime]

It doesn't make sense to talk of a plane in R^3 being contained in R^4 as if that were a well defined object. Even if we identify a copy of R^3 sitting inside R^4, any plane in R^3 is still a plane in R^4 so you just counted it when you specified the set of planes (passing through the origin). Also, R^2 contains only one 2-dimensional subspace - itself.

Doing it in coords, isn't the set of things (x,y,z,0) a subspace of R^4? What is its dimension?

 

[HallsofIvy]

So I have planes in R^4. But I also have planes in R^2 and R^3 passing thru (0,0,0,0) that are contained in R^4. Lastly, R^4, itself.

Once again what do you mean by "planes"? How is a "plane in R^2" different from a "plane in R^3" or a "plane in R^4"?