Write the three 3 by 3 matrices for 180 ° rotations about the x , y , z axes. Show that these three matrices commute (contrary to what we usually expect-see Problems 7.30 and 7.31). By writing the multiplication table, show that these three matrices with the unit matrix form a group. To which order 4 group is it isomorphic? Hint: See Problem 13.5.
Write the three 3 by 3 matrices for 180 ° rotations about the x , y , z axes. Show that these three matrices commute (contrary to what we usually expect-see Problems 7.30 and 7.31). By writing the multiplication table, show that these three matrices with the unit matrix form a group. To which order 4 group is it isomorphic? Hint: See Problem 13.5.
Write the three 3 by 3 matrices for
180
°
rotations about the
x
,
y
,
z
axes. Show that these three matrices commute (contrary to what we usually expect-see Problems 7.30 and 7.31). By writing the multiplication table, show that these three matrices with the unit matrix form a group. To which order 4 group is it isomorphic? Hint: See Problem 13.5.
Problem 3.
(1) Prove that (ac)(bd) = (abc)(abd) in Sn for distinct a, b, c, d ∈ {1, 2, 3, · · · , n}.
(2) Prove that the product of every pair of transpositions in Sn can be expressed as a product of at most two cycles of length 3.
(3) Prove that any permutation in the alternating group An with n ≥ 3 is a product of cycles of length 3.
Which of the following pairs (S, °) forms a group?
(d) S = R, ° is the usual multiplication of real numbers. (e) S= {[a b; c d] where a,b,c ∈ R.}. ° is the usual matrix multiplication.
A Problem Solving Approach to Mathematics for Elementary School Teachers (12th Edition)
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RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY