Page 18 of Springer texts of matrix algebra states, under the heading Unions & direct sums of vector spaces:
The vector space generated by the union of the sets in the individual vector spaces is easy to form since $(V,o)$ and $(W,o)$ are vector spaces, so for any vector $x$ in either $V$ or $W$, $ax$ is in that set.
How is this possible? Let us say $V=[2,4,7,5]$ and $W=[5,8,2,1]$. Let us take an element $5$ from $V$ and choose $a=10$. $ax$, in this case, will be $50$ which is not included in either $V$ or $W$.
Note that
$$V=[2,4,7,5]$$ and $$W=[5,8,2,1]$$ in your example are not vector spaces.
These are just vectors.
The vector space generated by those two linearly independent vectors is a two dimensional vector space which is made out of all possible linear combination of these two vectors.
If your scalar is $a=5$, then $ aV=5V = [10,20,35,25]$ is an element of your new vector space.