That is, the vector itself has a life of its own.

也就是说矢量本身是独立的
The components come in the minute you pick your axis.

但它的分量由你所选择的坐标系确定
It's not enough to say these are the components of the vector;

所以只说"这些是某矢量的分量"是不够的
you've got to tell me, "I am working with i and j,

你还得告诉我 "我选取的基矢是 i 和 j
which I define in the following manner."

它们是这样定义的"
I gave you a law of transformation of the components;

我介绍过分量变换的法则
namely, if the vector has components Ax and Ay in one reference frame

即如果矢量在一个坐标系的分量为 Ax 和 Ay
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