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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.

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  • $\begingroup$ Who says $\text{span}(S)=V$ in theorem $4.7$? $\endgroup$ – Mathematician 42 Mar 14 '17 at 18:13
  • $\begingroup$ Elementary Linear Algebra, Sixth Edition, By Larson, Edwards, and Falvo, chapter 4, section 4.4, Spanning Sets and Linear Independence, page# 211. It is not the book I am using but because I was confused so I decided to look it up and this is what I came across, it clarified most of the things but left me confused still. Also, it is not in the theorem but if you look at the definition of the span of a set, that is where it is stated. In the span theorem they are saying it is the smallest part of V. If span(s) = V then how is it the smallest? Because it should cover all of V. $\endgroup$ – Iamlearningmath Mar 14 '17 at 18:17
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    $\begingroup$ No, you're not reading it correctly. Definition of the span of set does not say that $\text{span}(S)=V$. It says that if $\text{span}(S)=V$ then we say that $V$ is spanned by $S$, but we don't ask this in theorem $4.7$. $\endgroup$ – Mathematician 42 Mar 14 '17 at 18:22
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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.

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