ℚ is the set of rational numbers? and \(\displaystyle m_1= \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\), \(\displaystyle m_2= \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}\), \(\displaystyle m_3= \begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}\), \(\displaystyle m_4= \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}\)?
So you are looking for a linear transformation, \(\displaystyle \psi_1\), from the space of all 2 by 2 matrices with rational entries such that \(\displaystyle \psi_1(m_1)\), \(\displaystyle \psi_1(m_2)\), \(\displaystyle \psi_1(m_3)\), and \(\displaystyle \psi_1(m_4)\) span the set of second degree polynomials with rational coefficients?
Since such a matrix can be written \(\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}\)\(\displaystyle = am_1+ bm_2+ cm_3+ dm_4\) where a, b, c, and d are rational numbers, one perfectly good answer would be the linear function \(\displaystyle \psi_1(a, b, c, d)= ax^2+ bx+ c\).
