July 13th, 2009, 06:56 AM  #1 
Newbie Joined: Oct 2007 Posts: 4 Thanks: 0  Standard matrix
Hello, 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 nonstandard 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? Erik 

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