isomorphism from one vector space to another one

Question

This is from my textbook

I don't quite understand what isomorphism means. Greek word "isomorphism" means same structure, but how can we say $P_3$ has the same structure as $R^4$?

Answer 1

An isomorphism between two objects should be a one-to-one, onto (hence invertible), and should preserve the structure of the objects in question.

When we consider vector spaces, the structure we care about is the vector space structure. You know the vector space structure is preserved if there is a mapping $$f : V \to W$$ such that $$f(a v_1 + b v_2) = a f(v_1) + b f(v_2)$$

Here, $\mathbb{P}_3$ has the same structure as $\mathbb{R}_4$ since you are taking each polynomial of the form $$ax^3 + bx^2 + cx + d$$ to the 4-tuple $$(a, b, c, d) \\ \text{where} \ a, b, c, d \in \mathbb{R}^4$$