Could someone please clarify for me the difference between a spanning list and a column space? Is it the same thing? The definitions that I have in front of me are as follows:
A list of vectors in a vector space is a spanning list of that vector space if every vector in the vector space can be written as a linear combination of the vectors in that list
The column space of a matrix $A$ is all the linear combinations of its column vectors, which is essentially the span of those vectors.
Also, I want to make sure I understand the difference between a vector space and a linear subspace. Once again, I am given two definitions and I want to know if these are interchangeable terms:
- A linear subspace contains the zero-vector, is closed under addition and scalar multiplication.
- A vector space is a nonempty set of vectors which is closed under the vector space operations.
I know a span is a linear subspace by the definition provided above, and of course by that, so is a column space. But are they also a vector space? Is a spanning list a vector space as well? Would appreciate if someone would help me get my terminology right here. Thanks!
A spanning list is not a vector space, but the set of linear combinations of vectors in the list is. For example, $\{(1,0),(0,1)$} is a spanning list for $\Bbb R^2$, but {$(1,0),(0,1)$} is not a vector space, since it does not contain $(2,0), (1,1)$, etc., though its span does.
A linear subspace is part of a larger vector space. A linear subspace is a vector space, but calling it a subspace emphasizes that it is part of a larger space, just as a subset is a set contained by a larger set.
The columns vectors of a matrix are the spanning list of the column space, but you could have a spanning list not in the context of matrix columns.