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 $A$)?
Best Answer
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 $b_1,\dots,b_n$."