Linear transformations defined on a basis

Let $T:\mathbf{P}_{2}\to\mathbb{R}$ be a linear transformation such that

Find $T(ax^{2}+bx+c)$.

Let vectors $\mathbf{v}_{1},\ldots,\mathbf{v}_{n}\in\mathbb{R}^{n}$ and $\mathbf{w}_{1},\ldots,\mathbf{w}_{n}\in\mathbb{R}^{m}$ be given. Let $A$ be the matrix whose columns are $\mathbf{v}_{1},\ldots,\mathbf{v}_{n}$, and assume that $A^{-1}$ exists. Show that there exists a linear transformation $T$ such that $T(\mathbf{v}_{i})=\mathbf{w}_{i}$ for $i=1,\ldots,n$.