Thread: Prrof: Subspace of a finite dimensional vector space is finite dimensional?Help pleas

Show that any subspace of a finite dimensional vector space is finite dimensional and state theorems used.

I would think that a subspace of a vector space would have fewer than or the same number of vectors in the basis as the vector space and must therefore be finite dimensional, but this does not use theorems just logic and I think my logic may be skewed. Does a subspace of a vector space have the same basis as the vector space and does this have anything to do with the proof? Can anyone help me prove this?

Any help would be appreciated. Thanks in advance.

Originally Posted by chocaholic

Show that any subspace of a finite dimensional vector space is finite dimensional and state theorems used.

I would think that a subspace of a vector space would have fewer than or the same number of vectors in the basis as the vector space and must therefore be finite dimensional, but this does not use theorems just logic and I think my logic may be skewed. Does a subspace of a vector space have the same basis as the vector space and does this have anything to do with the proof? Can anyone help me prove this?

Any help would be appreciated. Thanks in advance.

Do you know that in an -dimensional vector space a set of linearly independent vectors forms a basis for the space? Clearly the answer you are looking for falls out of this.

If you do not know the result, prove it. It isn't too hard. Just take a your linearly independent vectors, write them in terms of the basis vectors, and show that any of the basis vectors can be written in terms of these vectors.