Some confusion about the basis vectors. These questions came from the definition of a basis at the bottom. I'm wondering:
- What the relation is between basis and non-basis vectors. Or put another way, if every single element in the vector space is linearly dependent on the linearly independent basis vectors (they are independent among themselves, but dependent amongst the other vectors). This is based on my understanding of the definition of span and the idea that every vector in the space is a linear combination of the basis vectors (except the basis vectors). I'm visualizing the span as: the basis vectors are all linearly independent, but then they "connect" to every other vector outside of the basis via linear combination. In that sense they reach every other vector, and so span the vector space. Wondering if that is correct.
- If the number of basis vectors is the dimension. Just want to make sure I'm understanding.
- What the relation between a basis and a generating set is.
- A brief intuition on how to think of basis vectors.
A set of elements in a vector space is called a basis if they are:
- Linearly independent.
- And every vector in the vector space is a linear combination of the set (the basis vectors?).
In another perspective, a basis is a linearly independent spanning set.
Formally, a basis $B$ of a vector space $V$ over a field $F$ is a linearly independent subset of $V$ that spans $V$.
A span of a set of vectors in $V$ is the intersection of all subspaces containing that set.
In the definition of free module they compare a basis to a generating set.