let (a_1,a_2,...,a_n)^t be the coordinates of v in V, then f(v) = Av where f:V--> W
what does it mean that A is defined st f(v_j) = (sum from i=1 to m) for a_ij w_i
may i know how did v_j come about and hence what is a_ij?
im confused
The v_j surely make up a basis for V, and likewise the w_i make up a basis for W. The linear map f is determined completely by its values on a basis for V.
The matrix A is the matrix of f with respect to the chosen bases for V and W, and the (i,j)'th entry of A is just a_ij.
What do you mean by "transition matrix"? I've only encountered that term in finite Markov chain-theory.
The matrix A is the matrix of the linear map f with respect to the bases {v_j} and {w_i} of V and W, respectively.
This means that if v is a vector in V, and if (a_1,a_2,...,a_n)^t are the coordinates of v with respect to the basis {v_j} (which means that v = a_1v_1+a_2v_2+...+a_nv_n), then
$\displaystyle f(v) = A(a_1,a_2,\ldots,a_n)^t.$
In words, f and A do the exact same thing to the vector v, except that before you apply A to v, you need the coordinates of v with respect to the basis of V.
(Note that a lot of stuff has dangerously been named a_something).
Your question is confusing to me, but let me try to answer what I think you're asking. Correct me, if I'm misinterpreting you.
You suppose that f(v) = w. Then you write v in the basis {v_1,v_2} for V, that is v = b_1v_1+b_2v_2. You then have that
$\displaystyle w=f(v) = \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix}\begin{bmatrix}b_1 \\ b_2 \\ \end{bmatrix}=\begin{bmatrix}a_{11}b_1+a_{12}b_2 \\ a_{21}b_1+a_{22}b_2 \\ \end{bmatrix}.$
Note that this need to be interpreted correctly. The result on the right hand side is really the coordinates of w with respect to the basis {w_1,w_2}. This is because we can only construct a matrix from a linear map, when we have given bases for the two spaces V and W, and when dealing with such a matrix, it takes as input coordinates with respect to the basis of V, and gives as output coordinates with respect to the basis of W.
In other words, $\displaystyle (a_{11}b_1+a_{12}b_2,a_{21}b_1+a_{22}b_2)$ are the coordinates of w with respect to the basis (w_1,w_2), and hence
$\displaystyle w = (a_{11}b_1+a_{12}b_2)w_1 + (a_{21}b_1+a_{22}b_2)w_2.$