How do I
build a basis for the vector space $L(\Bbb R^3,\Bbb R_3[x])$? This is the vector space of all linear transformations that goes from $\Bbb R^3$ to the space of polynomials of degree 3 or less over $\Bbb R$
I know a basis for $\Bbb R^3$ and $\Bbb R_3[x]$, but since what I'm being asked is basis for functions how do I make linear combinations of functions to build another one?
Thanks for any help you can provide.
Since $\dim \mathbb R^3=3$ and $\dim\mathbb R_3[x]=4$, we may write the elements of $L(\mathbb R^3,\mathbb R_3[x])$ as $4\times 3$ matrices. If $p\in\mathbb R_3[x]$, let $p_n$ be the coefficient of $x^n$ in $p$. Let $\{e_1,e_2,e_3\}$ be the canonical basis of $\mathbb R^3$. For $T\in L(\mathbb R^3,\mathbb R_3[x])$, define $$M_T = \begin{bmatrix}T(e_1)_0&T(e_2)_0&T(e_3)_0\\ T(e_1)_1&T(e_2)_1&T(e_3)_1\\T(e_1)_2&T(e_2)_2&T(e_3)_2\\T(e_1)_3&T(e_2)_3&T(e_3)_3\\\end{bmatrix}. $$ Let $A_{ij}$ be the $4\times 3$ matrix with $(i,j)$ entry equal to $1$ and other entries equal to $0$. Then we may write $M_T$ uniquely as a linear combination of the $A_{ij}$ by $$M_T = \sum_{i=0}^3\sum_{j=1}^3 T(e_j)_iA_{ij}, $$ so the linear maps represented by the matrices $A_{ij}$ form a basis for $L(\mathbb R^3,\mathbb R_3[x])$.
