 # How to determine the span of two vectors in RR^2: (4,2) and (1,3) Do I subtract them? Heergerneuu 2022-09-16 Answered
How to determine the span of two vectors in ${\mathbb{R}}^{2}$
(4,2) and (1,3)
Do I subtract them? I don't how I'd solve this. Thanks in advance. In my question the vectors are like this:
$\left[\begin{array}{c}4\\ 2\end{array}\right]$
But that doesn't matter, right?
Would the vector equation ${x}_{1}{v}_{1}+{x}_{2}{v}_{2}$ = b be consistent for any b in ${\mathbb{R}}^{2}$?
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The span is just the possible linear combinations of the two vectors...
$Span\left\{\left(4,2\right),\left(1,3\right)\right\}=\left\{a\left(4,2\right)+b\left(1,3\right);a,b\in \mathbb{R}\right\}$
###### Not exactly what you’re looking for? trkalo84
The span of a set of vectors, is the set of every linear combination that you can "create" from those vectors.
So in your example a(4,2)+b(1,3), where $a,b\in \mathbb{R}$
So for example (5,5) is in the span of your vectors, because $1\cdot \left(4,2\right)+1\cdot \left(1,3\right)=\left(5,5\right)$
Also (3,−1) is in the span as $\left(4,2\right)-\left(1,3\right)=\left(3,-1\right)$
In general every vector of the form $\left(4a+b,2a+3b\right)$ are in the span.