Q:

Confused about the relation between linear transformations, matrices and basis vectors

I was watching 3blue1brown's video series on linear algebra.  My understanding till now is :-

A linear transformation takes in a vector and outputs another vector. 
The above statement is equivalent to multiplying a unique matrix to the given vector. 
3b1b shows the linear transformation using a new coordinate system, and shows that $\hat{i}$ and $\hat{j}$ change. 
When he discusses change of basis, he states that it helps us move between different coordinate systems. 
3b1b also states that a matrix implicitly assumes coordinate systems, as it represents the landing spots of basis vectors after linear transformation. 
He shows how to transform a rotation matrix in a conventional Cartesian coordinate system, to Jennifer's coordinate system (one where basis vectors are not perpendicular to one another). 

Points 4,5 and 6 have really confused me and now I doubt even points 1,2 and 3. 
When we write a matrix what basis vectors does it assume?  I have never seen any text stating that this assumes a Cartesian coordinate system.  I always assumed that it is somehow independent of coordinate systems.  
My second question: I thought that a linear transformation doing a 90° counter-clockwise rotation is represented by a unique matrix \begin{equation*}
A = 
\begin{pmatrix}
0 & -1  \\
1 & 0
\end{pmatrix},
\end{equation*}
but, as was shown in the video for Jennifer's choice of basis vectors the same 90° counter-clockwise rotation linear transformation is in fact \begin{equation*}
B = 
\begin{pmatrix}
1/3 & -2/3  \\
5/3 & -1/3
\end{pmatrix}. 
\end{equation*}
It seems like a linear transformation has a one-one mapping to a unique matrix only for a given set of basis vectors.  Thus, the same matrix can refer to different linear transformations if we choose a different basis vector.  In case, I am correct, could you provide a mathematically rigorous way of writing this down (using math symbols).  I feel that I understand concepts better if I can write it in a mathematical form, instead of relying solely on intuition.  

A:

A vector is an element of a vector space.  An element of a vector space can be an $n$-tuple of numbers, a polynomial, a matrix, a function etc. 
A linear transformation transforms a vector ($n$-tuple, polynomial, matrix, function, etc.)  into another vector ($n$-tuple, polynomial, matrix, function, etc.).  A matrix cannot transform a vector into another vector, because you can multiply a matrix by an $n$-tuple, but you can't multiply a matrix by a polynomial, a matrix (well, not always, see below), a function, etc. 
A matrix associated to a linear transformation can only multiply $n$-tuples of coordinates respect to a basis, and the results are $n$-tuples of coordinates respect to a basis. 
Imagine that your vector space is the set of all symmetric $2\times 2$ matrices, and that your linear transformation is:$$T\left(\begin{bmatrix} a & b \\ b & c \end{bmatrix}\right)=\begin{bmatrix} c & a \\ a & b \end{bmatrix}$$
The simplest basis is: $\left\{\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\right\}$.
Respect to this basis the coordinates of $\begin{bmatrix} a & b \\ b & c \end{bmatrix}$ are $(a,b,c)$, the coordinates of $\begin{bmatrix} c & a \\ a & b \end{bmatrix}$ are $(c,a,b)$.
The matrix associated to $T$ respect to that basis is: $\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$.
You can't multiply $\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$ by $\begin{bmatrix} a & b \\ b & c \end{bmatrix}$, but:
$$\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} a \\ b \\ c \end{bmatrix}=\begin{bmatrix} c \\ a \\ b \end{bmatrix}$$
i.e.
$$\begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\text{Coord}\left(\begin{bmatrix} a & b \\ b & c \end{bmatrix}\right)=\text{Coord}\left(\begin{bmatrix} c & a \\ a & b \end{bmatrix}\right)$$
This is why:

you always need a basis to associate a matrix to a linear
transformation (when the basis is omitted you assume the canonical basis),
the matrix associated to a linear transformation is unique respect to
a fixed basis,
you can also have different bases for the domain and the range of a
linear transformation, so the matrix associated to a linear transformation is unique respect to the basis of its domain and the basis of its range,
since there are infinite bases, there also are infinite matrices
associated to a linear transformation.