Change of basis of the vector

So I understand that the change of basis of the matrix is $X' = U^{\dagger}XU$. But how about if I just want to change the basis of a vector $v$.

Is $v' = U^{\dagger}v$ the right one? Why does the $U$ disappear here?

• You are using a lot of symbols without letting us know what any of them mean. – Gerry Myerson Sep 23 '16 at 7:18
• – Ka-Wa Yip Sep 23 '16 at 7:33
• Presumably $U$ is unitary and $\dagger$ is the adjoint. – Omnomnomnom Sep 23 '16 at 11:49
• The key is to understand that $X'v'$ should produce the same result as $Xv$, but that output $Xv$ of $X$ should be expressed with respect to the new basis. – Omnomnomnom Sep 23 '16 at 11:53
• @Omnomnomnom thank you. does that mean $v' = X'^{\dagger}Xv$? – Ka-Wa Yip Oct 6 '16 at 22:06