In the past I have thought of all matrices as linear transformations that transform space. And to a degree this is true. However, certain transformations of objects have identical counter parts in terms of transformation of space instead. This is seen clearly in [Relationship between Translating Space and Translating Objects](Relationship%20between%20Translating%20Space%20and%20Translating%20Objects.md). We can think of this cleanly as follows. If we have a vector and wish to rotate it 90 degrees, there are two ways to accomplish that: 1. Rotate the vector 90 degrees, leaving space ("gridlines", as defined by our basis vectors) fixed 2. Rotate space (our gridlines, as defined by our basis vectors) and leave our vector fixed in place These two approaches are isomorphic and identical. ### Active vs Passive Transformations A great blog post [here](http://www.michaelkeutel.de/blog/rotation-matrices-vector-basis/) outlines this well. See the image below:  > **Active Transformation** > $v$ is defined with respect to the standard basis, $E$. The rotation matrix then rotates $v$ to $v^\prime$, where $v^\prime$ is also described with respect to the standard basis. > > **Passive Transformation** > $v$ is defined in respect to the *rotated* basis, $B_R$. This means that $v$ is rotated in the same way that the basis $B_R$ is rotated in respect to the standard basis. The second step does the basis change and transforms the coordinates of $v$ from basis $B_R$ to the standard basis $E$. ### The main source of confusion The strange thing about the passive transformation is that we assume that $v$ is defined in respect to $B_R$ rather than the standard basis, and in the end the rotation finally yields $v$ in respect to the standard basis. To be even more clear: > The confusion arises because we *always*, naturally, think about vectors being described with respect to the *standard basis*. It is hard to break that association. However, that is exactly what we must do in this scenario. Note the following! In the earlier image, look at the active and passive sides. On the active side $v$ is *different* from $v$ on the passive side (they are literally *different* vectors; simply look at their angles. We know that $E$ and $B_R$ are different basis with different basis vectors, yet, the coefficient of $v$ and $[v]_{B_R}$ will be the same. Hence, they are certainly different vectors). And that is the *point*! In the passive case there are two distinct steps: 1. *Assume* that $v$ is described with respect to $B_R$ (this is the confusing part since we always assume a vector is described wrt the standard basis). This will *move* $v$ in the abstract space. 2. Rotate the coordinate system (i.e. $B_R$) so that it is mapped to the standard basis, $E$. ### A shortcoming of notation In reality I believe that most of the problem stems from the following: > We almost always **implicitly** assume we are working with the standard basis. This makes it very hard to see it any other way. Even an $n$-tuple such as $(4, 1, 7, 1)$ could have a *basis* of anything! It could mean $4$ apples, $1$ pear, $7$ oranges, and $1$ grape; as long as we have a notion of addition and scalar multiplication its allowed. In reality, **space** has no *intrinsic* coordinate system (no intrinsic “grid” or way to describe things). There are just **objects**. We then *define* how these objects can be represented in terms of other objects. A basis gives us a great way to do that. It always us to say that all things in our space can be composed by a combination of our basis objects. ### Whiteboard image (see [here](https://photos.google.com/photo/AF1QipOOgUa5MqTmfliunj94JJqB8lXDMp-TYTMTGDiD))  --- Date: 20220228 Links to: [Change of Basis Linear Algebra](Change%20of%20Basis%20Linear%20Algebra.md) [Relationship between Translating Space and Translating Objects](Relationship%20between%20Translating%20Space%20and%20Translating%20Objects.md) [Big Ideas MOC](Big%20Ideas%20MOC.md) Tags: References: * [Google Photos](https://photos.google.com/photo/AF1QipOOgUa5MqTmfliunj94JJqB8lXDMp-TYTMTGDiD) * Notability: Ways to view a matrix * Notability: Matrix Transpose and the Four Fundamental Subspaces * [Rotation matrices and vector basis change : Michael Keutel | Portfolio](http://www.michaelkeutel.de/blog/rotation-matrices-vector-basis/)