What is the difference between the span of a set to its subspace?

Question

I am confused with some of the definitions of linear algebra. I know that the span of set S is basically the set of all the linear combinations of the vectors in S.

The subspace of the set S is the set of all the vectors in S that are closed under addition and multiplication (and the zero vector).

So my question is, what is the difference between the span of S to the subspace of S? it seems as if you could find its span in its subspace and its subspace in its span.

Answer 1

A subspace $s$ of $S$ is a space within $S$. We have $4$ main subspaces, for instance. The most famous one is the linear combination of columns. For example, if we have $[C_1]$ and $[C_2]$ as columns of a matrix, we define column subspace as $a C_1 + b C_2$.

Any other subspace is just using the very simple idea of above. You can predict how the row space would be.

Now about span, the $a C_1 + b C_2$ is called the span of $C_1$ and $C_2$.

Answer 2

Take a nonempty subset $S$ of some vector space $V$. Then $\operatorname{span} S$ can be defined in two ways:

• $\operatorname{span} S$ is the set of all linear combinations of vectors in $S$.
• $\operatorname{span} S$ is the smallest subspace of $V$ that contains all the elements of $S$. (How do you construct $\operatorname{span} S$? Take the intersection of all subspaces of $V$ that contain all the element of $S$.)

It turns out that these definitions are equivalent, so if you take one as the definition of $\operatorname{span} S$, then you can prove the other bullet point above as a theorem. See, for instance, Section 2.2 of Hoffman and Kunze's book Linear Algebra, second edition.