Write the four rotation matrices for rotations of
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- Use the standard matrix for counterclockwise rotation in R2 to rotate the triangle with vertices (3,5), (5,3) and (3,0) counterclockwise 90 about the origin. Graph the triangles.arrow_forward2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .arrow_forwardFor each of the following matrices A in SL(2,R), list the elements of A and give the order | A |. a. A=[ 0111 ] b. A=[ 1110 ]arrow_forward
- Show that the matrix below does not have an LU factorization. A=0110arrow_forwardThe elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of section 3.1. Find the order of each element of the group. (Sec. 3.1,35) A permutation matrix is a matrix that can be obtained from an identity matrix In by interchanging the rows one or more times (that is, by permuting the rows). For n=3 the permutation matrices are I3 and the five matrices. (Sec. 3.3,22c,32c, Sec. 3.4,5, Sec. 4.2,6) P1=[ 100001010 ] P2=[ 010100001 ] P3=[ 010001100 ] P4=[ 001010100 ] P5=[ 001100010 ] Given that G={ I3,P1,P2,P3,P4,P5 } is a group of order 6 with respect to matrix multiplication, write out a multiplication table for G.arrow_forward15. Repeat Exercise with, the multiplicative group of matrices in Exercise of Section. 14. Let be the multiplicative group of matrices in Exercise of Section, let under multiplication, and define by a. Assume that is an epimorphism, and find the elements of. b. Write out the distinct elements of. c. Let be the isomorphism described in the proof of Theorem, and write out the values of.arrow_forward
- Find the centralizer for each element a in each of the following groups. The quaternion group G={ 1,i,j,k,1,i,j,k } in Exercise 34 of section 3.1 (Sec. 3.1, #34). G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1 (Sec. 3.1, #36). G={ I3,P1,P2,P3,P4,P5 } in Exercise 35 of section 3.1 (Sec. 3.1, #35). Sec. 3.1,34 34. Let G be the set of eight elements G={ 1,i,j,k,1,i,j,k } with identity element 1 and noncommutative multiplication given by (1)2=1, i2=j2=k2=1, ij=ji=k jk=kj=i, ki=ik=j, x=(1)x=x(1) for all x in G (The circular order of multiplication is indicated by the diagram in Figure 3.8.) Given that G is a group of order 8, write out the multiplication table for G. This group is known as the quaternion group. (Sec. 3.3,22a,32a, Sec. 3.4,2, Sec. 3.5,11, Sec. 4.2,8, Sec. 4.4,23, Sec. 4.5,40a, Sec. 4.6,3,11,16) Sec. 3.1,36 Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G. (Sec. 3.3,22b,32b, Sec. 4.1,22, Sec. 4.6,14) Sec. 3.1,35 35. A permutation matrix is a matrix that can be obtained from an identity matrix In by interchanging the rows one or more times (that is, by permuting the rows). For n=3 the permutation matrices are I3 and the five matrices. (Sec. 3.3,22c,32c, Sec. 3.4,5, Sec. 4.2,6) P1=[ 100001010 ] P2=[ 010100001 ] P3=[ 010001100 ] P4=[ 001010100 ] P5=[ 001100010 ] Given that G={ I3,P1,P2,P3,P4,P5 } is a group of order 6 with respect to matrix multiplication, write out a multiplication table for G.arrow_forwardFind all subgroups of the quaternion group.arrow_forward
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