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Math 250a Problem Set 2 Solutions

Math 250a Problem Set 2 Solutions - Math 250 Fall 2006...

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Math 250 – Fall 2006 Selected Solutions, Assignment #2 HH 1.7.16 c): Interpret the mapping A A 2 on 2 × 2 matrices as a mapping on R 4 by identifying a matrix a b c d with the column vector a b c d . Compute the derivative of the mapping on R 4 , and show that it agrees with the transport to R 4 of the mapping B AB + BA . Response: We compute that a b c d 2 = a 2 + bc ab + bd ca + dc dcb + d 2 = a 2 + bc b ( a + d ) c ( a + d ) bc + d 2 . The transfer of this to R 4 is ˜ sq : a b c d a 2 + bc b ( a + d ) c ( a + d ) bc + d 2 . If we differentiate ˜ sq according to the standard recipe (i.e., form the Jacobian matrix of partial derivatives), we find D ˜ sq = 2 a c b 0 b a + d 0 b c 0 a + d c 0 c b 2 d . On the other hand. if B = x y z w , the map B AB + BA is expressible as x y z w ax + bz ay + bw cx + dz cy + dw + xa + yc xb + yd za + wc zb + wd = 2 ax + bz + cy bx + ( a + d ) y + bw cx + ( a + d ) z + cw cy + bz + 2 wd . If we transfer this map to R 4 , we get x y z w 2 ax + cy + bz bx + ( a + d ) y + bw cx + ( a + d ) z + cw cy + bz + 2 wd = 2 a c b 0 b a + d 0 b c 0 a + d c 0 c b 2 d x y z w . Thus, we see that we get the same linear map for the derivative, whether we first transfer the squaring mapping to R 4 and then compute the derivative, or first compute the derivative intrinsically on the 2 × 2 matrices, and then transfer the result to R 4 . AS2.2: Which of the functions f j , j = 1, 2, 3, 4, are differentiable at the origin?
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