Math 250a Problem Set 2 Solutions

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 b a + d b c a + d c 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 b a + d b c a + d c 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 ....
View Full Document

This note was uploaded on 07/18/2008 for the course MATH 250 taught by Professor Rogerhowe during the Fall '06 term at Yale.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online