Nontrivial Subspace with Equal Dimensions: Linear Algebra Example

[dan0]

Hi,
I'm just learning for my linear algebra exam and I wonder if somebody could give me an example of a nontrivial subspace which has as many dimensions as the original space.
Thanks a lot

 

[Muzza]

What does non-trivial mean here? With "my" usage of non-trivial, I'd say it's not possible. Because if V is a finite-dimensional vector space and W is a subspace of V with dim(W) = dim(V), then V = W (this uses the fact that n linearly independent vectors in an n-dimensional space must necessarily form a basis of the space). Maybe it's possible with a space of infinite dimension, though.

 

[mathwonk]

yep. take polynomials of even degree, as a subspace of all polynomials. does that work?