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Is it true that, if A={u1,u2,u3} and B={v1,v2,v3} are two bases for a 3-dimensional vector space V, then P A->B=[[v1]A[v2]A[v3]A] where [vi]A denotes the coordinates

Is it true that, if A={u1,u2,u3} and B={v1,v2,v3} are two bases for a 3-dimensional vector space V, then P A->B=[[v1]A[v2]A[v3]A] where [vi]A denotes the coordinates of vector vi relative to the basis A

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