# Change of Basis ### Intuition regarding applying change of basis matrix to a vector Consider two coordinate systems, 1 and 2. Let us have a vector, that is described in coordinate system 1. Let us state that we have: * **Forward transformation**, $F$, that transforms our basis vectors of CS 1 to the locations of CS 2 (where they are still *described* in CS 1) * **Backward transformation**, $B$, where our basis vectors (of CS 2) are transformed to the location of our basis vectors in CS 1 (again, they are still described in CS 2). Now, if we are given the vector $\vec{v}$, described in CS 1, and we want to describe it in CS 2, how would we do that? Well, even thought it seems counter intuitive at first, we need to transform it via $B$! $B(\vec{v})$ Wait, why is that? Okay, well let's consider the following. If we want to describe our vector, below, in terms of the red CS (which is twice as large as the blue CS), we need to scale our vector down by one half:  Likewise, if we rotate our basis, note that our vector does not move. This means that from the perspective of the vector, the effect was the rotate *in the opposite/inverse direction*!  The key idea here is: > Vectors are **invariant**. To *ensure this invariance*, when our basis changes, in order to describe a vector in the new basis, *the inverse* transformation must be applied! This is needed to guarantee the invariance, otherwise we may suddenly be transforming the actual vector (scaling or rotating it), instead of simply changing its description. A note worth mentioning here: This certainly has many links to [Principle of Relativity](Principle%20of%20Relativity.md) and [Reference Frames](Reference%20Frames.md). Now, if we transform our basis via $F$, how do we ensure that we describe $\vec{v}$ correctly in a way the ensures invariance? We need to use $F^{-1}$, which is simply $B$. And that is why we need to transform $\vec{v}$ via $B$! --- References * [Tensors for Beginners 1: Forward and Backward Transformations](https://www.youtube.com/watch?v=sdCmW5N1LW4&list=PLJHszsWbB6hrkmmq57lX8BV-o-YIOFsiG&index=3)