I need a bit of clarification in regards to the spanning set. I am confused between the definition and the theorem.
Definition of Spanning Set of a Vector Space: Let $S = \{v_1, v_2,…v_n\}$ be a subset of a vector space $V$. The set is called a spanning
set of $V$ if every vector in $V$ can be written as a linear combination of vectors in $S$. In such cases it is said that $S$ spans $V$.
Definition of the span of a set: If $S = \{v_1, v_2,…v_n\}$ is a set of vectors in a vector space $V$, then the span of $S$ is the
set of all linear combinations of the vectors in $S$,
$span(S) = \{k_1v_1 + k_2v_2+…+k_nv_n | k_1, k_2,…k_n \in \mathbb{R}\}$.
The span of is denoted by $span(S)$ or $ span\{v_1, v_2,…v_k\}$. If $span(S) = V$ it is said that $V$ is spanned by $\{v_1, v_2,…v_n\}$, or that $S$ spans $V$.
What I understand from the definitions:
$S$ is a subset of the vector space $V$ and if I can represent all of the vectors that are in the vector space by using just the subset or the smaller part of $V$ then it can be said that $S$ spans $V$ or can reach every vector in $V$.
Linear combination has the following form $a = k_1v_1 + k_2v_2 + k_3v_3 +…+k_nv_n$ where $k_i$ are scalars and $v_i$ are the vectors in the subset $S$ of $V$ and $a$ is a particular vector in $V$ that can be created by a linear combination of vectors in $S$. This can be done for infinite number of vectors or all the vectors that are in the vector space $V$. We can create a set of all linear combinations of the vectors the can be reached by $S$ in $V$. For instance linear combination $a$ can be in the set and just like it, many others are a part of this set. We say that $S$ spans $V$ if every vector in $V$ can be reached by the vectors in $S$. Furthermore, $span(S)$ is the set that contains the linear combinations.
Theorem 4.7 Span(S) is a subspace of V: If $S = \{v_1, v_2,…v_n\}$ is a set of vectors in a vector space $V$. then $span(S)$ is a subspace
of $V$. Moreover, $span(S)$ is the smallest subspace of $V$ that contains $S$, in the sense that every other subspace of $V$ that contains $S$ must contain $span(S)$.
Question: Theorem 4.7 is where I am confused. The reason why I posted my understanding of the above definitions is so that if I am missing something perhaps someone will point it out to me so I can bridge the gap. Regardless, where I am confused is that the theorem states that $span(S)$ is the smallest part of $V$, but how can it be the smallest if we are saying that $span(S) = V$ in the definition of the span of a set. Should this not mean that the $span(S)$ is $V$ because of the equality? I can see that subset $S$ could be the smallest part because we are only taking the elements that can span $V$ and that will make sense, but $span(S)$ is supposed to be a set of linear combination and therefore contains every thing that is in $V$. What am I missing here?
P.S. Sorry for the long post, I have just been grappling with this for a while so I wanted to clarify. Also, I am self-studying so forums like these are my teachers.
Best Answer
The definition does not assume $\textrm{span}(S) = V.$ If this happens to be the case, $S$ is called a spanning set, but Theorem 4.7 does not make this assumption. In the theorem, $S$ is just any subset of $V.$ Consider for example $S = \{0\},$ in which case $\textrm{span}(S)$ is also just $\{0\}.$ Or consider $\{(1,0)\} \subset \mathbb{R}^2,$ whose span is the $x$-axis inside of the plane.