Let $\mathcal{P}_2$ denote the vector space of all polynomials in the variable $x$ of degree less than or equal to $2$. Let $\mathcal{B} = \lbrace 1, x, x^2 \rbrace$ be an ordered basis for $\mathcal{P}_2$. Let $\lbrack ~ \rbrack_{\mathcal{B}} : \mathcal{P}_2 \to \mathbb{R}^3$ be the linear transformation determined by $$\lbrack 1 \rbrack_{\mathcal{B}} = \vec{e}_1, \ \ \ \lbrack x \rbrack_{\mathcal{B}} = \vec{e}_2, \ \ \ \lbrack x^2 \rbrack_{\mathcal{B}} = \vec{e}_3.$$ Find the coordinate vector representation for each of the following polynomials. Your answers should be vectors of the general form $\verb+<1,2,3>+$.

1. $\lbrack -6 \rbrack_{\mathcal{B}} =$ .
   
   
2. $\lbrack 1-6x^{2} \rbrack_{\mathcal{B}} =$ .
   
   
3. $\lbrack 5x^{2}+9x+1 \rbrack_{\mathcal{B}} =$ .
   
   
4. Is the linear transformation $\lbrack ~ \rbrack_{\mathcal{B}}$ an isomorphism? choose isomorphism not an isomorphism