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# Change of basis matrix

nosplit "Change of basis" words: 152
The change of basis matrix is the matrix that allows us to express the new basis in an old basis: $$xold​=Cxnew​ (52)$$
Mnemonic is as follows: consider we have an initial basis . Now, we define the new basis in terms of the old basis, e.g.: $$xnew​ynew​​=1xold​+2yold​=3xold​+4yold​​ (53)$$ which can be written in matrix form as: $$[xnew​ynew​​]=[13​​24​][xold​yold​​] (54)$$ and so if we set: $$M=[13​​24​] (55)$$ we have: $$xnew​​=Mxold​​ (56)$$
The usual question then is: given a vector in the new basis, how do we represent it in the old basis?
The answer is that we simply have to calculate the matrix inverse of : $$xold​​=M−1xnew​​ (57)$$
That is the matrix inverse.