Suppose a change of basis from basis ##B## to basis ##C## is represented by the matrix ##S##.(adsbygoogle = window.adsbygoogle || []).push({});

That is, ##S## is the transformation matrix from ##B## to ##C##.

Now if ##t## is a given linear transformation, ##t:~V\rightarrow V##, with eigenvectors ##\epsilon_i##, say, and ##T## is the representation of ##t## in ##B##, then, the representation of t in ##C## is ##STS^{-1}##.

Now, if the representation of ##\epsilon_i## in basis ##B## be ##v^B_i##, then the representation of ##\epsilon_i## in basis ##C##, ##v^C_i=Sv^B_i##.

But shouldn't the vectors themselves transform in an opposite sense to the transformation of the basis, that is, shouldn't it be that ##v^C_i=S^{-1}v^B_i## ? I'm getting really confused. Please, can someone clarify?

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# Change of Basis

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