Let be a subspace of the vector space generated by and , and be a subspace generated by and . Show the dimension of the following subspaces: , , and give a basis for each.

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- Nov 12th 2011, 11:05 AMnorickaDimension and basis
Let be a subspace of the vector space generated by and , and be a subspace generated by and . Show the dimension of the following subspaces: , , and give a basis for each.

- Nov 12th 2011, 11:44 AMDrexel28Re: Dimension and basis
- Nov 12th 2011, 12:23 PMnorickaRe: Dimension and basis
- Nov 12th 2011, 01:18 PMDevenoRe: Dimension and basis
how many basis elements do M and N have?

- Nov 12th 2011, 01:40 PMnorickaRe: Dimension and basis
Both of them have 3? (Happy)

- Nov 12th 2011, 02:01 PMDevenoRe: Dimension and basis
how can a set generated by 2 elements have 3 basis vectors???

to be fair, p1 and p2 are only listed as generating elements, its not explicitly stated whether or not {p1,p2} forms a basis.

but, by the very definition of "generate" they are elements of a span-set. are they linearly independent? (if you decide they are, how can you be sure?

and what does this mean dim(M) is?) - Nov 12th 2011, 02:13 PMnorickaRe: Dimension and basis
p1 and p2 are linearly independent, so their dimension is 2?

- Nov 12th 2011, 02:16 PMDevenoRe: Dimension and basis
elements do not have dimension, vector spaces (and their subspaces) have dimension.

the dimension of a vector space, V, is defined to be the number of elements in ANY basis.

if S is a subset of V, then dim(span(S)) ≤ |S|.

if S is a linearly independent set, then dim(span(S)) = |S|. - Nov 12th 2011, 02:29 PMnorickaRe: Dimension and basis
So the dimension of the subspace M generated by p1, and p2 is 2, then. And that's the same with N as well.

- Nov 12th 2011, 02:37 PMnorickaRe: Dimension and basis
Was my idea of the dimension of M+N right? And what about the bases?