Definition for Basis of a Subspace

Question

As I gradually work through the fundamentals of linear algebra, I have often found myself struggling with boiling down the concepts that I have learned to a concise, all encompassing definition which can be summed up in no more than a couple of sentences.

I am currently learning about bases for vector spaces, and for this I seek clarification of my definition from Stack Exchange members.

From what I have learned so far, the concisest definition that I can provide for this concept is the following:

A basis of a subspace is a set of vectors which can be used to represent any other vector in the subspace.

Thus the set must:

• Be linearly independent.
• Span all of the subspace.
• Not include any vectors which are linearly dependent upon other vectors in the set.

Is this definition accurate? If not; where did I misspeak? And is there any crucial information that I missed?

Answer 1

Your first and third conditions assert the same thing. Therefore, I would keep just the first (it is the shortest one) and the second one. If you want to be more concise, you can say that a basis of a vector space is a linearly independet spanning subset of that space.

Answer 2

"Thus" in your statement doesn't actually include any justification. The reason that you want the basis vectors to be linearly independent is so that each vector in the subspace can be uniquely represented as a linear combination of the basis vectors.