# How to Define the Display Matrix from two linear mappings

I'm working right now on this task, but i dont't know how to start, Im glad for every hint, I hope we can solve this task together $$Given \, are \, the \, following \, ordered \, Bases \, of \, \mathbb{R²}, \mathbb{R³} \, and \, \mathbb{R⁴}$$ $$B₂= \begin{pmatrix} 1 \\ 0 \end{pmatrix} , \begin{pmatrix} 0 \\ 1 \end{pmatrix} \quad B₃= \begin{pmatrix} 1 \\ 1 \\ 1 \\ \end{pmatrix} , \begin{pmatrix} 1 \\ 1 \\ 0 \\ \end{pmatrix} , \begin{pmatrix} 1 \\ 0 \\ 0 \\ \end{pmatrix} \quad B₄= \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \\ \end{pmatrix} , \begin{pmatrix} 1 \\ 1 \\ 1 \\ 0 \\ \end{pmatrix} , \begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \\\end{pmatrix} , \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \\\end{pmatrix}$$ $$Let \, us \, consider \, now \, two \, linear \, mappings \, \varphi:\mathbb{R²} → \mathbb{R³} \, \, \psi: \mathbb{R³} \, → \, \mathbb{R⁴}$$ $$\varphi \begin{pmatrix} v1 \\ v2 \end{pmatrix} = \begin{pmatrix} v1 - v2 \\ 0 \\ 2v1 - v2 \end{pmatrix} \quad and \, \psi \begin{pmatrix} v1\\ v2 \\ v3 \\ \end{pmatrix} = \begin{pmatrix} v₁ + 2v₃ \\ v₂ - v₃ \\ v₁ + v₂ \\ 2v₁ + 3v₃ \\ \end{pmatrix}$$ $$Define \, the \, display \, matrix \, M \begin{matrix} B₂ \\ B₃ \end{matrix} (\varphi), \, M \begin{matrix} B₃ \\ B₄ \end{matrix} (\psi) \, and \, M \begin{matrix} B₂ \\ B₄ \end{matrix} (\psi \, ○ \, \varphi)$$

Thank you for any hint :) Definition of the display matrix

• Do you know the definition of a display matrix? – user635162 Jan 26 at 17:24
• Yes sure, I added it as a picture in the main question. – dean Jan 26 at 18:02
• Then just apply it. Put each basis vector into the function and write the corresponding image as a linear combination of the other basis. The coefficients of that linear combination are then the entries of the matrix, just as the definition states. – user635162 Jan 26 at 18:08