# Show that the vector space of polynomials R[x] is isomorphic to a proper subspace of itself

Show that the vector space of polynomials R[x] is isomorphic to a proper subspace of itself:

Vector Space Isomorphism exists when there exists a bijective (one-to-one and onto) linear mapping F:V $\rightarrow$U. the coefficient of the polynomials can be written as $(a_0,a_1,a_2...)$. But how to find the subspace?

What about instead to prove the dimension of the the two vector spaces is the same, which means isomorphic? But how to do it?

• Proper subspace means a subspace that is not $\Bbb R[x]$. – Arnaud D. May 11 '17 at 10:20
• Hint: consider $p(x) \mapsto x p(x)$. – Kenny Wong May 11 '17 at 10:28
• If it helps, you might want to forget the fact they're polynomials, and just think of them as sequences $(a_0, a_1, \ldots)$ where each $a_i \in R$. There is nothing polynomial-specific being used here. – Joppy May 11 '17 at 13:26

First, note that you can't do this with finite dimensional vector spaces. You have to come up with a linear transformation which is one-to-one but not onto. Any linear transformation is determined by what happens to the vectors in any basis. Use the standard basis for $R[x]$, that is, $x^n$ where $n\ge 0$. You have to map each $x^n$ to a polynomial such that the mapping is one-to-one. One way to do this is to map $x^n$ to a polynomial of degree $n+1$. You can check this ensures that only the zero polynomial maps to zero. You can check that it is not onto because, for example, $x^0$ is not the image of any polynomial. These two facts depend on looking at the leading term of polynomials and seeing that no nonzero polynomial is a linear combination of polynomials of lesser degrees.
Hint: Consider $p(x) \mapsto p(x^2)$.
This acts on a sequence of coefficients (which is all a polynomial is) by inserting zeros between them: $$(a_0,a_1,a_2,\dots,a_n,0,0,0,\dots) \mapsto (a_0,0,a_1,0,a_2,\dots,a_{n-1},0,a_n,0,0,0,\dots)$$ and so you can recover one from the other.