1. coordinates

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

2. 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.

3. does the matrix A form the transition matrix from W to V?

4. 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

$f(v) = A(a_1,a_2,\ldots,a_n)^t.$

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).

5. Isint f(v) =w? Does that mean in your case that if v= b_1v_1 +b_2 v_2 then f(v) = a_i1 b_1 + a_i2 b_2 =w , where i is the rows of A(2x2 matrix)

i assumed that the basis of V is {v_1, v_2}

6. 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

$w=f(v) = \begin{bmatrix}a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix}\begin{bmatrix}b_1 \\ b_2 \\ \end{bmatrix}=\begin{bmatrix}a_{11}b_1+a_{12}b_2 \\ a_{21}b_1+a_{22}b_2 \\ \end{bmatrix}.$

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, $(a_{11}b_1+a_{12}b_2,a_{21}b_1+a_{22}b_2)$ are the coordinates of w with respect to the basis (w_1,w_2), and hence

$w = (a_{11}b_1+a_{12}b_2)w_1 + (a_{21}b_1+a_{22}b_2)w_2.$

7. yes this is what im asking. so in this case, f(v)= w where w is not the image but actually the coordinate vector wrt to the image right?

8. Yes, (a_11b_1+a_12b_2 , a_21b_1+a_22b_2) is the coordinates of w with respect to the basis (w_1,w_2).