Change of basis: trying to understand tensors

I've been trying to understand tensors. I have learned that they are a way of doing maths in higher dimensions - that is, going beyond the 3 dimensional world with which we are familiar. The 2D and 3D worlds are subsumed within this framework. And axes do not have to be perpendicular to each other. I've been struggling with that so last night I decided to investigate what happens in 2D. An important thing about the world of tensors is that a given vector must remain unchanged, no matter what set of coordinate axes we are using. They seem to use  the word BASIS to describe the axis set up. So I think my picture above shows a change of basis. The red vector remains the same in each basis but the components of the vector, shown in green, change. I'm working on this because I want to understand the significance of contravariant and covariant factors. I need examples to help my thinking. This post is thinking out loud. They'll be more when I've made progress.