Can a Nontrivial Subspace Have the Same Dimension as Its Original Space?

[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

 

[HallsofIvy]

What do you mean by "nontrivial". I would call a subspace nontrivial as long as it was not just the 0 vector. If that's the case then just take any vectors space V with V as the subspace.

Another possibility, a lot more non-trivial would be an infinite dimensional vector space with an infinite dimensional subspace.
The vector space of all polynomials with subspace the space of all even polynomials comes to mind.

 

[thepatientmental]

!

Sure, no problem! One example of a nontrivial subspace with the same dimension as the original space is the set of all linear combinations of two linearly independent vectors in a three-dimensional space. This subspace would have a dimension of 2, the same as the original three-dimensional space. Another example could be the set of all symmetric matrices in a space of square matrices, which would have a dimension equal to the number of variables in the original space. I hope this helps with your exam preparation! Good luck!