and Ax prime and Ay prime in another reference frame,

在另一坐标系中的分量为 Ax' 与 Ay'
how are the two related?

它们有着什么样的联系
They come from writing i prime and j prime in terms of i and j,

写出用 i 和 j 来表示 i' 和 j' 的式子
then sticking it in an expression

再代入分量的表达式
and identifying the new representation.

就能得到新的式子
Somehow, when I told you to invert the transformation,

我上次让你们逆推变换公式的时候
some of you had some difficulty.

有的同学觉得很困难
Maybe you didn't realize

也许你并没有意识到
that they are just simultaneous equations that you solve.

它们其实就是你们平时解的方程组
Normally, if I tell you 3x + 4y = 6,

一般来说如果我给你两个方程 3x + 4y = 6
and 9x + 6y = 14, you know how to solve it.

和 9x + 6y = 14 你们肯定会解
It's like really the same kind of problem,

这其实是一样的问题