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# la9 - THE TRANSPOSE Definition The transpose of an m...

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THE TRANSPOSE Definition. The transpose of an m × n -matrix A , is the n × m -matrix A T which exchanges rows for columns. If A has entries a ij then A T has entries a ji . Example. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 T = 1 6 11 2 7 12 3 8 13 4 9 14 5 10 15 Proposition. The dot product is v · w = v T w Proof. v T w = v 1 . . . v n T w 1 . . . w n = ( v 1 · · · v n ) w 1 . . . w n = v 1 w 1 + · · · + v n w n = v · w Proposition. ( A T ) T = A Proof. Exchange rows for columns. Then do it again. You get back what you started with. Example. 1 2 3 4 5 6 T ! T = 1 4 2 5 3 6 T = 1 2 3 4 5 6 Proposition. ( AB ) T = B T A T Proof. Remember that the i, j -entry of AB is the dot product of the i th row of A and the j th column of B . Thus the i, j -entry of ( AB ) T is the dot product of the j th row of A and the i th column of B . Thus the i, j -entry of ( AB ) T is the dot product of j th column of A T and the i th row of B T . By symmetry of the dot

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