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07-matrix - MATRIX PRODUCT HOMEWORK Section 2.3...

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10/6/2003, MATRIX PRODUCT Math 21b, O. Knill HOMEWORK: Section 2.3: 10,20,30,40*,42*, Section 2.4: 14,28,40*,72* MATRIX PRODUCT. If B is a p × m matrix and A is a m × n matrix, then BA is defined as the p × n matrix with entries ( BA ) ij = m k =1 B ik A kj . EXAMPLE. If B is a 3 × 4 matrix, and A is a 4 × 2 matrix then BA is a 3 × 2 matrix. B = 1 3 5 7 3 1 8 1 1 0 9 2 , A = 1 3 3 1 1 0 0 1 , BA = 1 3 5 7 3 1 8 1 1 0 9 2 1 3 3 1 1 0 0 1 = 15 13 14 11 10 5 . COMPOSING LINEAR TRANSFORMATIONS. If S : R n R m , x 7→ Ax and T : R m R p , y 7→ By are linear transformations, then their composition T S : x 7→ B ( A ( x )) is a linear transformation from R n to R p . The corresponding n × p matrix is the matrix product BA . EXAMPLE. Find the matrix which is a composition of a rotation around the x -axes by an agle π/ 2 followed by a rotation around the z -axes by an angle π/ 2. SOLUTION. The first transformation has the property that e 1 e 1 , e 2 e 3 , e 3 → - e 2 , the second e 1 e 2 , e 2 → - e 1 , e 3 e 3 . If A is the matrix belonging to the first transformation and B the second, then BA is the matrix to the composition. The composition maps e 1 → - e 2 e 3 e 1 is a rotation around a long diagonal. B = 0 - 1 0 1 0 0 0 0 1 A = 1 0 0 0 0 - 1 0 1 0 , BA =
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