Suppose that $(V, +, \cdot)$ is a vector space over a field $F$ and $S = \{v_1, v_2, \ldots, v_k\}$ is a subset of $V$. Describe the span of $S$. Explain how to determine whether $S$ is linearly independent.
This is a past exam question and it's not worth many marks because its just an explanation question but I was wondering if someone could provide me with a good definition to both parts. Thanks
The span of a set of vectors is the collection of all linear combinations of those vectors. Another way of writing this is
$$ Span(S) \;\; =\;\; \{a_1v_1 + \ldots + a_kv_k \; | \; a_1, \ldots, a_k \in F\}. $$
To determine whether a set of vectors is linearly independent you need to prove that if $a_1v_1 + \ldots + a_k v_k = 0$ then all of the coefficients must be zero, namely $a_1 = \ldots = a_k = 0$.