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I am asking this question to clarify some properties of a spanning set and subspaces. My textbook says the following:

If set $S$ spans the vector space $V$, then it can be said that $S$ is the spanning set of $V$.

If $S$ is a set of vectors in vector space $V$, then the span of $S$ is the set of all linear combination of the vectors of $S$ and is a subspace of the vector space.

My question is

  1. if set $S$ spans the entire vector space $V$, then can the following also be inferred?
  • set $S$ is a subspace of vector space $V$;
  • vector space $V$ is a subspace of $S$.
  1. My textbook claims that a vector space $V$ is a subspace of $V$ itself. If S spans vector space $V$ and is a subspace of $V$, does this make $V$ equivalent or equal to set $S$ of vectors given that they are both vector spaces that can also be subspaces of each other??
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  • $\begingroup$ $S$ need not be a vector space, in general, so you cannot conclude that $S$ is a subspace of $V$, or vice versa. What you can say is that $\langle S \rangle$, the subspace generated by $S$, is equal to $V$. $\endgroup$
    – user169852
    Oct 23, 2018 at 18:54
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    $\begingroup$ To take an example, let $S = \{(1, 0), (0, 1)\}$. This set spans $\mathbb R^2$ but it is not a subspace of $\mathbb R^2$, because it isn't a vector space. But we do have $\langle S \rangle = \mathbb R^2$. $\endgroup$
    – user169852
    Oct 23, 2018 at 18:55
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    $\begingroup$ Why isn't the S considered a vector space? Which axioms does it fail? $\endgroup$
    – Evan Kim
    Oct 23, 2018 at 19:03
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    $\begingroup$ It doesn't contain the zero vector $(0,0)$. It isn't closed under addition, e.g. it doesn't contain $(1,0) + (0,1) = (1,1)$. It isn't closed under scalar multiplication, e.g. it doesn't contain $2(1,0) = (2,0)$. $\endgroup$
    – user169852
    Oct 23, 2018 at 20:56
  • $\begingroup$ ahhh okay. That makes more sense. I am trying to visualize the set in my head....is (0,1) and (1,0) points on the 2d plane? How can you change the set to form a line on the 2d plane? $\endgroup$
    – Evan Kim
    Oct 24, 2018 at 3:10

2 Answers 2

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No, $S$ does not need to be a vector space at all. For example, in the vector space $V = \mathbb R^2$, $S$ might consist of just two vectors.

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  1. No. For the simple reason that a spanning set is NOT (in general) a vector space, nor a subspace of a vector space.

  2. Hence this question is meaningless.

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