Linear Combination, Span, Generating Set.
$V/F$ and $S$ a non-empty subset of $V$. A vector $v \in V$ is called a linear combination of vectors from $S$ if there exists FINITE many vectors $v_1, v_2, …, v_n \in S$ and $a_1, a_2, …, a_n \in F$ s.t. $$v = a_1v_1 + a_2v_2 + … + a_nv_n$$ Furthermore, in this case, we say that $v$ is a linear combination of $v_1, v_2, …, v_n$ with coefficients $a_1, a_2, …, a_n$.
$V/F, S$ is a non-empty subset of $V$. The span of $S$ or the linear span of $S$ denoted by $\spa{S}$ is the set of all possible linear combination of vectors of $S$.
We define $\spa{\emptyset} = \set{0}$.
$\abs S$ could be finite or infinite.
$V/F$. If $S \subseteq V$, then $\spa{S} \leq V$.
Moreover, if $W \leq V, S\subseteq W$, then $\spa{S} \leq W$.
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$V/F$. If $S \subseteq V$, then $\spa{S} \leq V$ and $\spa{S}$ is the smallest subspace containing $S$.
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Let $V$ be a vector space over a field $F$. a subset $S \subseteq V$ is called a generating set if $\spa{S} = V$.
In this case we say $S$ generates $V$.
$S= \set{(1,0), (0,1)}$, then $S$ generates $\R^2$.
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Prove that $T=\set{(1, -1), (2, 3)}$ is a generating set.
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$S= \set{1, x-x^2, x^2, x^3+x^2}$
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$V / F$,
$$\spa{S}=V, S \subseteq T, T \subseteq V \implies \spa{T}=V$$
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