Q:

Finding the change in basis vectors from a linear transformation

At about $4:00$ my question pertaining to 3Blue1Brown's video of a collection of linear algebra questions begins. The video is here.

After a linear transformation, the vector changes, but so do the "boxes" in the "graph" of the vector space, if that makes sense. Before the transformation, the boxes in the vector space looked pretty standard, and then after the transformation, they seemed to increase in scale. If you keep your eye on the box where the tail of the vector $-1 \hat i + 2\hat j $ lands, you can see it's stretched and rotated to be far larger. Thus, there is a new "graph" or coordinate plane we're working in. But why then, does the narrator classify the new basis vectors in terms of the dulled out, original coordinate plane, with the original, standard-oriented boxes, calling transformed $\hat i$ $\hat i -2 \hat j$ when it's still a basis vector, so it shouldn't be a $\hat j$ in it to begin with? It should just be the new $\hat i$ with $1$ as its length in $x$ and $0$ as its length in $y$. 
Also, all of these kinds of transformations imply that our vectors start in the identity matrix form, so I'm assuming something like a "Shear" can only be classified if our $\hat i$ and $\hat j$ matrices are in the traditional orientation.

A:

I think you're getting confused as to what exactly a linear transformation does.  Remember that $T$ takes a vector (either a point with respect to the objective, "dull" grid or a pair of coordinates) and produces an output vector.  So in the context of the video, $T(\hat i) = \hat i - 2\hat j$.  Note that $\hat i$ refers to the specific vector $(1,0)$ in 2-dimensional space; it does not simply mean "the first vector of our basis".  While it is true that the vectors $T(\hat i)$ and $T(\hat j)$ form a new basis (for the particular transformation illustrated), those vectors are different from the inputs $\hat i$ and $\hat j$.  It does not make sense to say that $T(\hat i)$ "is the new $\hat i$".
For your second point, I have no idea what the statement "these kinds of transformations imply that our vectors start in the identity matrix form" is supposed to mean, but it makes no sense as stated.

