Home | 18.022 | Chapter 16 | Section 16.1

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The transformation A takes the jth column basis vector into the jth column of the matrix A; which is a linear combination of the basis vectors, with coefficient Akj for the basis vector indexed by k.
B takes the kth basis vector to the kth column of B, and takes this linear combination of basis vectors to the same linear combination of the columns of B.
Thus the entry (BA)ij consists of the sum of the Bik, each weighted by Akj: