I'm more frequently coming across phrases such as "vector $b$ spanned by $\{b_1, \dots , b_n\}$" and "$A$ spans $B$" while studying linear algebra. What do the terms "spanned by" and "spans" mean in this context?

For example: Does "$A$ spans $B$" mean that span$(A)$ = $B$ (where span() is the function whose output is the span of set $A$)?

  • $\begingroup$ It means we can find linear combinations of the vectors in the spanning set to get any vector in the spanned vector space. $\endgroup$ – mathreadler Aug 25 '17 at 15:48
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    $\begingroup$ Yes, "A spans B" means span(A)=B. I don't think I've ever called an individual vector spanned by a set though - instead I'd say the vector is in the span. $\endgroup$ – anon Aug 25 '17 at 15:49
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    $\begingroup$ I guess your textbook explains such a basic concept. $\endgroup$ – Francesco Polizzi Aug 25 '17 at 15:49
  • $\begingroup$ @Francesco Polizzi You guess wrong. I'm not using a textbook at the moment. $\endgroup$ – Aluthren Aug 25 '17 at 15:54
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    $\begingroup$ @Aluthren: so you are studying from some random sources? Bad move. Anyway, that's not my business. $\endgroup$ – Francesco Polizzi Aug 25 '17 at 16:00

If $V$ is a vector space, and $A$ is a subset of $V$, and $W$ is a vector subspace of $V$, then the phrase "$A$ spans $W$" means that each vector in $W$ can be written as a linear combination of vectors from $A$. Stated succinctly, $A$ spans $W$ if $\operatorname{span}(A) = W$, where $$ \operatorname{span}(A) = \big\{\sum_{\text{finite}}\alpha_iv_i\bigm| \text{$\alpha_i$ is a scalar, and $v_i\in A$}\big\}. $$

You will also hear "$W$ is spanned by $A$" if $A$ spans $W$. You will not hear phrases like "The vector $b$ is spanned by vectors $b_1,\dots,b_n$," since it is vector spaces that are spanned, not individual vectors. Instead, you may hear something like "The vector $b$ lies in the span of the vectors $b_1,\dots,b_n$."

  • $\begingroup$ Thank you for the clarification. $\endgroup$ – Aluthren Aug 25 '17 at 16:14

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