$$ P=a_0+a_1x+a_2x^2+\cdots+a_nx^n\\ V=a_1v_1+a_2v_2+a_3v_3+\cdots+a_nv_n $$ Why is Polynomial infinite dimensional here, but the linear combination is not?
Or are there any difference between these two expressions based on both have the similar form.
Thanks
The reason that the space of polynomials is considered infinite dimensional is that while a single polynomial can only have finitely many terms, each of finite degree, there is no bound on how many terms it can have, or which degree those terms can have.
In your example, you've stopped the polynomial at degree $n$, and if we can only use up to and including degree $n$, then polynomials become a finite-dimensional vector space (of dimension $n+1$, to be specific). But without any such restriction, the infinitely many polynomials $1, x, x^2, x^3, \ldots$ are all linearly independent, and as such the vector space is infinite dimensional.
There is nothing stopping you from doing the same thing with "regular" vectors, i.e. lists of numbers. You can allow addition of two vectors with different number of entries by padding the shortest one with $0$'s on the right until they have the same length, and then add them together the usual way. Also, we may requirre that any trailing zeroes on the right side of a vector are cut (except the final zero if all entries are zero). So, for instance $$ (1, 2, 3) + (1, 0, 0, 1, 2) = (1, 2, 3, 0, 0) + (1, 0, 0, 1, 2) = (2, 2, 3, 1, 2)\\ (1, 2, 3) + (5, 4, -3) = (6, 6, 0) = (6, 6) $$ (scaling is done the normal way). This will make a vector space that is entirely analoguous to the space of polynomials, only expressed the way you are used to seeing vectors. It also has the geometric interpretation of including $\Bbb R$ as the $x$-axis in $\Bbb R^2$, including $\Bbb R^2$ as the $xy$-plane in $\Bbb R^3$, and so on all the way up through the dimensions, and looking at the final union of all these vector spaces as a single vector space.
This is, in some sense, the smallest infinite-dimensional vector space you can make. Unlike many other, "larger" infinite-dimensional vector spaces, this one actually has a basis even without invoking Zorn's lemma. That means we can actually know what the elements of the basis are.