Thread: Subspace of R3

Hello,

I think I have the answer to this, but I want to be sure. Even though it's an odd-numbered problem, the answer to it is not in the back of the book (more theoretical answers are omitted.)

Problem:
Give a geometrical description of all subspaces of R^3. Justify your answer.

Proposed Solution:

All subspaces of R^3 could be
(1) all of R^3
(2) planes through the origin
(3) lines through the origin
(4) the zero vector (origin)

Justification:
The basis of R^3 could have either 3, 2, 1 or 0 dimensions. With 3 dimensions, the subspace would be all of R^3. With 2 dimensions, a plane can be formed with the two basis vectors and must go through the origin to satisfy the zero vector property of subspaces. With 1 dimension, a line will be formed that must go through the origin for the same reason as above. Finally, the zero vector will be the subspace if there are no basis vectors (0 dimensions), which satisfies all properties of a subspace.

Am I on the right track?

Thanks.

your line of reasoning is correct, but your statement "The basis of R^3 could have either 3, 2, 1 or 0 dimensions" is incorrect.

a basis does not have dimension, dimension (of a vector space) is the number of basis elements. and any basis for R^3 has 3 elements.

i think what you mean to say, is that a basis for a subspace of R^3 could have 0,1,2 or 3 elements.

Ah, yes, thank you. I meant that, for example, if the problem as basis {u1, u2, u3}, then the dimension is 3. If the basis is {u1, u2} the dimension is 2, etc...

Correct?