If you haven't seen it, it might be worth checking out 3Blue1Brown's Linear Algebra YouTube series (https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ

If you haven't seen it, it might be worth checking out 3Blue1Brown's Linear Algebra YouTube series (https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQ...). I think it's exactly that kind of "intuition of what Linear algebra is and is for".