What is a necessary condition to form a set of basis?

A collection of vectors is said to span the space if every vector can be written as a linear combination of the members of this set. A set of linearly independent vectors that span the space is called basis How can a set of linearly independent vectors span the space if necessary condition of spanning is that every vector can be written as the linear combination of the members which makes it linearly dependent??

• This is not a contradiction, it only means that the unique way to write an element $b$ of the basis is to write it as $b=b$. – Jesko Hüttenhain Dec 17 '17 at 10:05

The vectors from the basis are linearly independent. None of them is a linear combination of the other vectors from the basis.

The vectors from the basis span the whole space. Every other vector in the vector space is a linear combination of the vectors in the basis.

I don't see any contradiction here.

• so, that means a set of linearly independent vectors can span the space?? – akshit sharma Dec 17 '17 at 10:31
• Yes, they certainly can. I would also like to show you an example, but not sure which vector space is the closest to your mind. Take, for example, the vectors in geometry, in a plane. Pick any two non-zero and non-collinear vectors $v$ and $w$. They would make a basis. None of them is a linear combination of the other one, and every vector $u$ in the plane can be written as $u=\alpha v+\beta w$ for some $\alpha,\beta\in\mathbb R$. – user491874 Dec 17 '17 at 10:36
• okay, got it .thanks – akshit sharma Dec 17 '17 at 10:40

Who said that in a linear combination of vectors of a set of vectors, all vectors in the set are actually involved?

It is even necessary that all vectors of a spanning set do not appear in the linear combination in the case of an infinite dimensional vector space, such as the ring of polynomials $K[X]$ over a field $K$.

What the notion of basis adds to the notion of spanning set (or set of generators) is simply the fact that the linear combination that decomposes a given vector is unique.

A set of $n$ vectors are a basis for a space or a subspace with dimension $n$ if and only if they are linearly independent.

Conversely if in a space or a subspace you can find at most $n$ vectors linearly independent we define $n$ the dimension of the space.