Understanding Span and Linear Independence in Vector Spaces

[darkchild]

My professor says that a linearly independent subset of a vector space automatically spans the vector space, and that a subset of a vector space that spans the vector space is automatically linearly independent.

I don't understand why either of these is true.

 

[HallsofIvy]

they aren't. For example, a set containing a single non-zero vector is always independent but only spans in the case of a one-dimensional vector space. The set consisting of the entire vector space, on the other hand, spans the vector space but cannot be independent. You may have misunderstood your professor. A maximal linearly independent set (there is no linearly independent set containing more vectors) spans the space and a minimal spanning set (there is no smaller spanning set) is linearly independent. One can then show that all set that both span the space and are linearly independent contain the same number of vectors (the "dimension" of the space).