Vector space to $\mathbb{R^3}$ isomorphism

Question

The space of quadratic polynomials V with real coefficients.

Let $\alpha \in \mathbb{R}, u = u_{1}+u_{2}t+u_{3}t^{2}$ and $v=v_{1}+v_{2}t+v_{3}t^{2}$, then define $\alpha u := \alpha u_{1}+\alpha u_{2} t+\alpha u_{3}t^{2}$,

u+v is done pointwise.

I want to show that an isomorphism exists from V to $\mathbb{R}^{3}$.

However, this seems impossible. First off, I know I need to show a homomorphism from V to $\mathbb{R}^{3}$. Then, a bijection from V to $\mathbb{R}^{3}$.

Without any further information provided by the question, I define

$L:V\rightarrow \mathbb{R}^{3}$

$u \mapsto \left ( \bar{u}_{1},\bar{u}_{2},\bar{u}_{3} \right )$

which appears to be correct. But it seems as though the question is making an assumption that I must know of certain assumption to be made. I spent quite a while pulling my hair out over this question. Any hint is appreciated.

Thanks in advance.

Answer 1

Hint: two finite dimensional vector spaces $A$ and $B$ are isomorphic if and only if dim($A$) = dim($B$)

Answer 2

@saldukoo 2 finite dimensional vector spaces $A$ and $B$ are isomorphic if and only if $$wo vector spaces $A$ and $B$ are isomorphic if and only if $dim(A) = dim(B)$.