# Thread: Change of basis and linear maps?

1. ## Change of basis and linear maps?

If I have a linear map T:V -> W and the basis of V is B={v1,...,vn} and the basis of W is B={w1,...,wm}

Then what is the formula for the matrix M_AB(T)?

It may not make much sense, but in the previous section of my notes, the formula for changing coordinates when changing bases was x=C_ABy
where x is the coordinates/vectore in the B basis, y is the coordinates/vector in the A basis and C_AB is a matrix.

2. Originally Posted by supaman5
If I have a linear map T:V -> W and the basis of V is B={v1,...,vn} and the basis of W is B={w1,...,wm}

Then what is the formula for the matrix M_AB(T)?
The first column of the matrix corresponding to the the transformation T with respect to the bases in V and W is the coordinates of T(v1) with respect to the basis in W.

In other words, calculate T(v1), which will be a vector i W. Write this in terms of the basis in W, so T(v1) = c_{11}w1+c_{21}w2 + ... + c_{m1}wm, where the c_{i1} are scalars. Then the first column of the matrix consists of exactly these scalars.