Linear Algebra - Finding linear maps

[a12]

Find a linear map that satisfies the stated condition,

a linear map ψ_1: Mat_2x2(ℚ) → ℚ₂[t] such that ψ₁(m₁), ψ₁(m₂), ψ₁(m₃), ψ₁(m₄) span ℚ₂[t]

Thankyou!

 

[HallsofIvy]

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 $\displaystyle \psi_1(a, b, c, d)= ax^2+ bx+ c$.