Suppose we have a linear transformation such as T(v) = Mv.
And we also have T(v1)=u1, T(v2) = u2 ... T(vn) = un
Then we can find the coeficiants of the standard basis by b = (A^-1)v0 where A is the matrix such as [v1 v2 ... vn] and v0 is a vector of the standard basis.
with these coeficiants we can find the linera transformation of the standard basis such as Bi = b1u1 + b2u2 + ... + bnun
now the standard matrix is defined as [ B1 B2 ... Bn]
and this now satisfies the equation T(v) = Mv
eg: T(v1) = Mv1 is going to work.
My question is as follows:
Why doesnt the linear transformation T(v) = Mv work if M is obtained by using any non-standard basis of v? Why do we have to use the standard basis to find a matrix M that satisfies the linear transformation? Why doesnt any basis work?