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

In summary, the conversation discusses the concept of a nontrivial subspace with the same number of dimensions as the original space. One example given is an infinite dimensional vector space with an infinite dimensional subspace, such as the vector space of polynomials with the subspace of all even polynomials. The definition of a nontrivial subspace is also clarified.
  • #1
dan0
4
0
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
 
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  • #2
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.
 
  • #3
!

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!
 

1. What is a subspace?

A subspace is a subset of a vector space that satisfies all the properties of a vector space. This means that it is closed under vector addition and scalar multiplication, and contains the zero vector.

2. What is the dimension of a subspace?

The dimension of a subspace is the number of linearly independent vectors it contains. It represents the minimum number of vectors needed to span the subspace.

3. How is the dimension of a subspace related to the dimension of the vector space it is a part of?

The dimension of a subspace cannot be greater than the dimension of the vector space it is a part of. It can be equal to the dimension of the vector space if the subspace is the entire vector space.

4. How can the dimension of a subspace be determined?

The dimension of a subspace can be determined by finding a basis for the subspace. A basis is a set of linearly independent vectors that span the subspace. The number of vectors in the basis will be equal to the dimension of the subspace.

5. Can a subspace have a dimension of zero?

No, a subspace must have a dimension of at least one. This is because it must contain at least the zero vector, which is necessary for it to be a vector space. A subspace with a dimension of one is simply a line passing through the origin, while a subspace with a dimension greater than one can be a plane, a hyperplane, or even the entire vector space.

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