Editing Linear combination (section)

=== Euclidean vectors ===
Let the field ''K'' be the set '''R''' of [[real number]]s, and let the vector space ''V'' be the [[Euclidean space]] '''R'''<sup>3</sup>.
Consider the vectors {{nowrap|1='''e'''<sub>1</sub> = (1,0,0)}}, {{nowrap|1='''e'''<sub>2</sub> = (0,1,0)}} and {{nowrap|1='''e'''<sub>3</sub> = (0,0,1)}}.
Then ''any'' [[Euclidean vector|vector]] in '''R'''<sup>3</sup> is a linear combination of '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, and&nbsp;'''e'''<sub>3</sub>.

To see that this is so, take an arbitrary vector (''a''<sub>1</sub>,''a''<sub>2</sub>,''a''<sub>3</sub>) in '''R'''<sup>3</sup>, and write:
:<math>
\begin{align}
( a_1 , a_2 , a_3) & = ( a_1 ,0,0) + (0, a_2 ,0) + (0,0, a_3) \\[6pt]
& =  a_1 (1,0,0) + a_2 (0,1,0) + a_3 (0,0,1) \\[6pt]
& =  a_1 \mathbf e_1 +  a_2 \mathbf e_2 + a_3 \mathbf e_3.
\end{align}
</math>